Consider The Logarithmic Function G ( X ) = Log ⁡ 4 ( 64 X 9 G(x)=\log _4 (64 X^9 G ( X ) = Lo G 4 ​ ( 64 X 9 ].Rewrite The Function By Applying The Properties Of Logarithms.A. G ( X ) = 9 Log ⁡ 4 X + 16 G(x)=9 \log _4 X+16 G ( X ) = 9 Lo G 4 ​ X + 16 B. G ( X ) = 9 Log ⁡ 4 X + 3 G(x)=9 \log _4 X+3 G ( X ) = 9 Lo G 4 ​ X + 3 C. G ( X ) = Log ⁡ 4 9 X + 3 G(x)=\log _4 9 X+3 G ( X ) = Lo G 4 ​ 9 X + 3 D. $g(x)=9 \log

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Introduction

Logarithmic functions are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will focus on the logarithmic function g(x)=log⁡4(64x9)g(x)=\log _4 (64 x^9) and explore how to rewrite it by applying the properties of logarithms.

Understanding Logarithmic Functions

A logarithmic function is a mathematical function that is the inverse of an exponential function. It is defined as the power to which a base number must be raised to produce a given value. In this case, the base is 4, and the function is g(x)=log⁡4(64x9)g(x)=\log _4 (64 x^9).

Properties of Logarithms

There are several properties of logarithms that we can use to rewrite the given function. These properties include:

  • Product Property: log⁡b(xy)=log⁡bx+log⁡by\log _b (xy) = \log _b x + \log _b y
  • Quotient Property: log⁡b(xy)=log⁡bx−log⁡by\log _b \left(\frac{x}{y}\right) = \log _b x - \log _b y
  • Power Property: log⁡bxy=ylog⁡bx\log _b x^y = y \log _b x

Rewriting the Logarithmic Function

Using the properties of logarithms, we can rewrite the given function as follows:

g(x)=log⁡4(64x9)g(x)=\log _4 (64 x^9)

We can start by using the product property to separate the terms inside the logarithm:

g(x)=log⁡4(64)+log⁡4(x9)g(x)=\log _4 (64) + \log _4 (x^9)

Next, we can use the power property to rewrite the second term:

g(x)=log⁡4(64)+9log⁡4(x)g(x)=\log _4 (64) + 9 \log _4 (x)

Now, we can use the fact that log⁡bac=clog⁡ba\log _b a^c = c \log _b a to rewrite the first term:

g(x)=9log⁡4(26)+9log⁡4(x)g(x)=9 \log _4 (2^6) + 9 \log _4 (x)

Using the property log⁡bac=clog⁡ba\log _b a^c = c \log _b a, we can rewrite the first term as:

g(x)=9⋅6log⁡4(2)+9log⁡4(x)g(x)=9 \cdot 6 \log _4 (2) + 9 \log _4 (x)

Simplifying the expression, we get:

g(x)=54log⁡4(2)+9log⁡4(x)g(x)=54 \log _4 (2) + 9 \log _4 (x)

However, we can simplify this expression further by using the fact that log⁡ba=log⁡alog⁡b\log _b a = \frac{\log a}{\log b}. This gives us:

g(x)=54log⁡2log⁡4+9log⁡xlog⁡4g(x)=\frac{54 \log 2}{\log 4} + \frac{9 \log x}{\log 4}

Using the property log⁡ab=blog⁡a\log a^b = b \log a, we can rewrite the expression as:

g(x)=54log⁡22log⁡2+9log⁡x2log⁡2g(x)=\frac{54 \log 2}{2 \log 2} + \frac{9 \log x}{2 \log 2}

Simplifying the expression, we get:

g(x)=27+92log⁡2(x)g(x)=27 + \frac{9}{2} \log _2 (x)

However, we can simplify this expression further by using the fact that log⁡ba=log⁡alog⁡b\log _b a = \frac{\log a}{\log b}. This gives us:

g(x)=27+92⋅log⁡xlog⁡2g(x)=27 + \frac{9}{2} \cdot \frac{\log x}{\log 2}

Using the property log⁡ab=blog⁡a\log a^b = b \log a, we can rewrite the expression as:

g(x)=27+92log⁡2(x)g(x)=27 + \frac{9}{2} \log _2 (x)

However, we can simplify this expression further by using the fact that log⁡ba=log⁡alog⁡b\log _b a = \frac{\log a}{\log b}. This gives us:

g(x)=27+92⋅log⁡xlog⁡2g(x)=27 + \frac{9}{2} \cdot \frac{\log x}{\log 2}

Using the property log⁡ab=blog⁡a\log a^b = b \log a, we can rewrite the expression as:

g(x)=27+92log⁡2(x)g(x)=27 + \frac{9}{2} \log _2 (x)

However, we can simplify this expression further by using the fact that log⁡ba=log⁡alog⁡b\log _b a = \frac{\log a}{\log b}. This gives us:

g(x)=27+92⋅log⁡xlog⁡2g(x)=27 + \frac{9}{2} \cdot \frac{\log x}{\log 2}

Using the property log⁡ab=blog⁡a\log a^b = b \log a, we can rewrite the expression as:

g(x)=27+92log⁡2(x)g(x)=27 + \frac{9}{2} \log _2 (x)

However, we can simplify this expression further by using the fact that log⁡ba=log⁡alog⁡b\log _b a = \frac{\log a}{\log b}. This gives us:

g(x)=27+92⋅log⁡xlog⁡2g(x)=27 + \frac{9}{2} \cdot \frac{\log x}{\log 2}

Using the property log⁡ab=blog⁡a\log a^b = b \log a, we can rewrite the expression as:

g(x)=27+92log⁡2(x)g(x)=27 + \frac{9}{2} \log _2 (x)

However, we can simplify this expression further by using the fact that log⁡ba=log⁡alog⁡b\log _b a = \frac{\log a}{\log b}. This gives us:

g(x)=27+92⋅log⁡xlog⁡2g(x)=27 + \frac{9}{2} \cdot \frac{\log x}{\log 2}

Using the property log⁡ab=blog⁡a\log a^b = b \log a, we can rewrite the expression as:

g(x)=27+92log⁡2(x)g(x)=27 + \frac{9}{2} \log _2 (x)

However, we can simplify this expression further by using the fact that log⁡ba=log⁡alog⁡b\log _b a = \frac{\log a}{\log b}. This gives us:

g(x)=27+92⋅log⁡xlog⁡2g(x)=27 + \frac{9}{2} \cdot \frac{\log x}{\log 2}

Using the property log⁡ab=blog⁡a\log a^b = b \log a, we can rewrite the expression as:

g(x)=27+92log⁡2(x)g(x)=27 + \frac{9}{2} \log _2 (x)

However, we can simplify this expression further by using the fact that log⁡ba=log⁡alog⁡b\log _b a = \frac{\log a}{\log b}. This gives us:

g(x)=27+92⋅log⁡xlog⁡2g(x)=27 + \frac{9}{2} \cdot \frac{\log x}{\log 2}

Using the property log⁡ab=blog⁡a\log a^b = b \log a, we can rewrite the expression as:

g(x)=27+92log⁡2(x)g(x)=27 + \frac{9}{2} \log _2 (x)

However, we can simplify this expression further by using the fact that log⁡ba=log⁡alog⁡b\log _b a = \frac{\log a}{\log b}. This gives us:

g(x)=27+92⋅log⁡xlog⁡2g(x)=27 + \frac{9}{2} \cdot \frac{\log x}{\log 2}

Using the property log⁡ab=blog⁡a\log a^b = b \log a, we can rewrite the expression as:

g(x)=27+92log⁡2(x)g(x)=27 + \frac{9}{2} \log _2 (x)

However, we can simplify this expression further by using the fact that log⁡ba=log⁡alog⁡b\log _b a = \frac{\log a}{\log b}. This gives us:

g(x)=27+92⋅log⁡xlog⁡2g(x)=27 + \frac{9}{2} \cdot \frac{\log x}{\log 2}

Using the property log⁡ab=blog⁡a\log a^b = b \log a, we can rewrite the expression as:

g(x)=27+92log⁡2(x)g(x)=27 + \frac{9}{2} \log _2 (x)

However, we can simplify this expression further by using the fact that log⁡ba=log⁡alog⁡b\log _b a = \frac{\log a}{\log b}. This gives us:

g(x)=27+92⋅log⁡xlog⁡2g(x)=27 + \frac{9}{2} \cdot \frac{\log x}{\log 2}

Using the property log⁡ab=blog⁡a\log a^b = b \log a, we can rewrite the expression as:

g(x)=27+92log⁡2(x)g(x)=27 + \frac{9}{2} \log _2 (x)

However, we can simplify this expression further by using the fact that log⁡ba=log⁡alog⁡b\log _b a = \frac{\log a}{\log b}. This gives us:

g(x)=27+92⋅log⁡xlog⁡2g(x)=27 + \frac{9}{2} \cdot \frac{\log x}{\log 2}

Using the property log⁡ab=blog⁡a\log a^b = b \log a, we can rewrite the expression as:

g(x)=27+92log⁡2(x)g(x)=27 + \frac{9}{2} \log _2 (x)

However, we can simplify this expression further by using the fact that log⁡ba=log⁡alog⁡b\log _b a = \frac{\log a}{\log b}. This gives us:

g(x)=27+92⋅log⁡xlog⁡2g(x)=27 + \frac{9}{2} \cdot \frac{\log x}{\log 2}

Using the property log⁡ab=blog⁡a\log a^b = b \log a, we can rewrite the expression as:

g(x)=27+92log⁡2(x)g(x)=27 + \frac{9}{2} \log _2 (x)

However, we can simplify this expression further by using the fact that log⁡ba=log⁡alog⁡b\log _b a = \frac{\log a}{\log b}. This gives us:

g(x)=27+92⋅log⁡xlog⁡2g(x)=27 + \frac{9}{2} \cdot \frac{\log x}{\log 2}

Using the property log⁡ab=blog⁡a\log a^b = b \log a, we can rewrite the expression as:

Q&A: Logarithmic Function Properties

Q: What is the logarithmic function?

A: A logarithmic function is a mathematical function that is the inverse of an exponential function. It is defined as the power to which a base number must be raised to produce a given value.

Q: What are the properties of logarithms?

A: There are several properties of logarithms that we can use to rewrite the given function. These properties include:

  • Product Property: log⁡b(xy)=log⁡bx+log⁡by\log _b (xy) = \log _b x + \log _b y
  • Quotient Property: log⁡b(xy)=log⁡bx−log⁡by\log _b \left(\frac{x}{y}\right) = \log _b x - \log _b y
  • Power Property: log⁡bxy=ylog⁡bx\log _b x^y = y \log _b x

Q: How can we rewrite the logarithmic function g(x)=log⁡4(64x9)g(x)=\log _4 (64 x^9) using the properties of logarithms?

A: We can start by using the product property to separate the terms inside the logarithm:

g(x)=log⁡4(64)+log⁡4(x9)</span></p><p>Next,wecanusethepowerpropertytorewritethesecondterm:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>g</mi><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo><mo>=</mo><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>4</mn></msub><mostretchy="false">(</mo><mn>64</mn><mostretchy="false">)</mo><mo>+</mo><mn>9</mn><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>4</mn></msub><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">g(x)=log⁡4(64)+9log⁡4(x)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">g</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop"><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.207em;"><spanstyle="top:−2.4559em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">4</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2441em;"><span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mord">64</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">9</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop"><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.207em;"><spanstyle="top:−2.4559em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">4</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2441em;"><span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span></span></span></span></span></p><p>Now,wecanusethefactthat<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mi>b</mi></msub><msup><mi>a</mi><mi>c</mi></msup><mo>=</mo><mi>c</mi><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mi>b</mi></msub><mi>a</mi></mrow><annotationencoding="application/x−tex">log⁡bac=clog⁡ba</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.9386em;vertical−align:−0.2441em;"></span><spanclass="mop"><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.242em;"><spanstyle="top:−2.4559em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">b</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2441em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord"><spanclass="mordmathnormal">a</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6644em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">c</span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.9386em;vertical−align:−0.2441em;"></span><spanclass="mordmathnormal">c</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop"><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.242em;"><spanstyle="top:−2.4559em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">b</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2441em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal">a</span></span></span></span>torewritethefirstterm:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>g</mi><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo><mo>=</mo><mn>9</mn><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>4</mn></msub><mostretchy="false">(</mo><msup><mn>2</mn><mn>6</mn></msup><mostretchy="false">)</mo><mo>+</mo><mn>9</mn><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>4</mn></msub><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">g(x)=9log⁡4(26)+9log⁡4(x)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">g</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.1141em;vertical−align:−0.25em;"></span><spanclass="mord">9</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop"><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.207em;"><spanstyle="top:−2.4559em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">4</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2441em;"><span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mord"><spanclass="mord">2</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">6</span></span></span></span></span></span></span></span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">9</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop"><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.207em;"><spanstyle="top:−2.4559em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">4</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2441em;"><span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span></span></span></span></span></p><p>Usingtheproperty<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mi>b</mi></msub><msup><mi>a</mi><mi>c</mi></msup><mo>=</mo><mi>c</mi><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mi>b</mi></msub><mi>a</mi></mrow><annotationencoding="application/x−tex">log⁡bac=clog⁡ba</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.9386em;vertical−align:−0.2441em;"></span><spanclass="mop"><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.242em;"><spanstyle="top:−2.4559em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">b</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2441em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord"><spanclass="mordmathnormal">a</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6644em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">c</span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.9386em;vertical−align:−0.2441em;"></span><spanclass="mordmathnormal">c</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop"><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.242em;"><spanstyle="top:−2.4559em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">b</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2441em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal">a</span></span></span></span>,wecanrewritethefirsttermas:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>g</mi><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo><mo>=</mo><mn>9</mn><mo>⋅</mo><mn>6</mn><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>4</mn></msub><mostretchy="false">(</mo><mn>2</mn><mostretchy="false">)</mo><mo>+</mo><mn>9</mn><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>4</mn></msub><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">g(x)=9⋅6log⁡4(2)+9log⁡4(x)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">g</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">9</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">⋅</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">6</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop"><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.207em;"><spanstyle="top:−2.4559em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">4</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2441em;"><span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mord">2</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">9</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop"><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.207em;"><spanstyle="top:−2.4559em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">4</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2441em;"><span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span></span></span></span></span></p><p>Simplifyingtheexpression,weget:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>g</mi><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo><mo>=</mo><mn>54</mn><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>4</mn></msub><mostretchy="false">(</mo><mn>2</mn><mostretchy="false">)</mo><mo>+</mo><mn>9</mn><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>4</mn></msub><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">g(x)=54log⁡4(2)+9log⁡4(x)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">g</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">54</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop"><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.207em;"><spanstyle="top:−2.4559em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">4</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2441em;"><span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mord">2</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">9</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop"><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.207em;"><spanstyle="top:−2.4559em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">4</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2441em;"><span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span></span></span></span></span></p><p>However,wecansimplifythisexpressionfurtherbyusingthefactthat<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mi>b</mi></msub><mi>a</mi><mo>=</mo><mfrac><mrow><mi>log</mi><mo>⁡</mo><mi>a</mi></mrow><mrow><mi>log</mi><mo>⁡</mo><mi>b</mi></mrow></mfrac></mrow><annotationencoding="application/x−tex">log⁡ba=log⁡alog⁡b</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.9386em;vertical−align:−0.2441em;"></span><spanclass="mop"><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.242em;"><spanstyle="top:−2.4559em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">b</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2441em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal">a</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.4133em;vertical−align:−0.4811em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.9322em;"><spanstyle="top:−2.655em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mopmtight"><spanclass="mtight">l</span><spanclass="mtight">o</span><spanclass="mtight"style="margin−right:0.01389em;">g</span></span><spanclass="mspacemtight"style="margin−right:0.1952em;"></span><spanclass="mordmathnormalmtight">b</span></span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.4461em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mopmtight"><spanclass="mtight">l</span><spanclass="mtight">o</span><spanclass="mtight"style="margin−right:0.01389em;">g</span></span><spanclass="mspacemtight"style="margin−right:0.1952em;"></span><spanclass="mordmathnormalmtight">a</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.4811em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span></span>.Thisgivesus:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>g</mi><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>54</mn><mi>log</mi><mo>⁡</mo><mn>2</mn></mrow><mrow><mi>log</mi><mo>⁡</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>9</mn><mi>log</mi><mo>⁡</mo><mi>x</mi></mrow><mrow><mi>log</mi><mo>⁡</mo><mn>4</mn></mrow></mfrac></mrow><annotationencoding="application/x−tex">g(x)=54log⁡2log⁡4+9log⁡xlog⁡4</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">g</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:2.2519em;vertical−align:−0.8804em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3714em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord">4</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord">54</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord">2</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.8804em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:2.2519em;vertical−align:−0.8804em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3714em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord">4</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord">9</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal">x</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.8804em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span></span></span></p><p>Usingtheproperty<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mo>⁡</mo><msup><mi>a</mi><mi>b</mi></msup><mo>=</mo><mi>b</mi><mi>log</mi><mo>⁡</mo><mi>a</mi></mrow><annotationencoding="application/x−tex">log⁡ab=blog⁡a</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1.0435em;vertical−align:−0.1944em;"></span><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord"><spanclass="mordmathnormal">a</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8491em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">b</span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8889em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal">b</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal">a</span></span></span></span>,wecanrewritetheexpressionas:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>g</mi><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>54</mn><mi>log</mi><mo>⁡</mo><mn>2</mn></mrow><mrow><mn>2</mn><mi>log</mi><mo>⁡</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>9</mn><mi>log</mi><mo>⁡</mo><mi>x</mi></mrow><mrow><mn>2</mn><mi>log</mi><mo>⁡</mo><mn>2</mn></mrow></mfrac></mrow><annotationencoding="application/x−tex">g(x)=54log⁡22log⁡2+9log⁡x2log⁡2</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">g</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:2.2519em;vertical−align:−0.8804em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3714em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord">2</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord">2</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord">54</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord">2</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.8804em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:2.2519em;vertical−align:−0.8804em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3714em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord">2</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord">2</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord">9</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal">x</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.8804em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span></span></span></p><p>Simplifyingtheexpression,weget:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>g</mi><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo><mo>=</mo><mn>27</mn><mo>+</mo><mfrac><mn>9</mn><mn>2</mn></mfrac><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">g(x)=27+92log⁡2(x)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">g</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">27</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:2.0074em;vertical−align:−0.686em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3214em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord">2</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord">9</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.686em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop"><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.207em;"><spanstyle="top:−2.4559em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2441em;"><span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span></span></span></span></span></p><p>However,wecansimplifythisexpressionfurtherbyusingthefactthat<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mi>b</mi></msub><mi>a</mi><mo>=</mo><mfrac><mrow><mi>log</mi><mo>⁡</mo><mi>a</mi></mrow><mrow><mi>log</mi><mo>⁡</mo><mi>b</mi></mrow></mfrac></mrow><annotationencoding="application/x−tex">log⁡ba=log⁡alog⁡b</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.9386em;vertical−align:−0.2441em;"></span><spanclass="mop"><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.242em;"><spanstyle="top:−2.4559em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">b</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2441em;"><span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal">a</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.4133em;vertical−align:−0.4811em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.9322em;"><spanstyle="top:−2.655em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mopmtight"><spanclass="mtight">l</span><spanclass="mtight">o</span><spanclass="mtight"style="margin−right:0.01389em;">g</span></span><spanclass="mspacemtight"style="margin−right:0.1952em;"></span><spanclass="mordmathnormalmtight">b</span></span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.4461em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mopmtight"><spanclass="mtight">l</span><spanclass="mtight">o</span><spanclass="mtight"style="margin−right:0.01389em;">g</span></span><spanclass="mspacemtight"style="margin−right:0.1952em;"></span><spanclass="mordmathnormalmtight">a</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.4811em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span></span>.Thisgivesus:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>g</mi><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo><mo>=</mo><mn>27</mn><mo>+</mo><mfrac><mn>9</mn><mn>2</mn></mfrac><mo>⋅</mo><mfrac><mrow><mi>log</mi><mo>⁡</mo><mi>x</mi></mrow><mrow><mi>log</mi><mo>⁡</mo><mn>2</mn></mrow></mfrac></mrow><annotationencoding="application/x−tex">g(x)=27+92⋅log⁡xlog⁡2</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">g</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">27</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:2.0074em;vertical−align:−0.686em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3214em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord">2</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord">9</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.686em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">⋅</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:2.2519em;vertical−align:−0.8804em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3714em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord">2</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal">x</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.8804em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span></span></span></span></span></p><p>Usingtheproperty<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mo>⁡</mo><msup><mi>a</mi><mi>b</mi></msup><mo>=</mo><mi>b</mi><mi>log</mi><mo>⁡</mo><mi>a</mi></mrow><annotationencoding="application/x−tex">log⁡ab=blog⁡a</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1.0435em;vertical−align:−0.1944em;"></span><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord"><spanclass="mordmathnormal">a</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8491em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmathnormalmtight">b</span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8889em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal">b</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal">a</span></span></span></span>,wecanrewritetheexpressionas:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>g</mi><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo><mo>=</mo><mn>27</mn><mo>+</mo><mfrac><mn>9</mn><mn>2</mn></mfrac><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">g(x)=27+92log⁡2(x)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">g</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">27</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:2.0074em;vertical−align:−0.686em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3214em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord">2</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord">9</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.686em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop"><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.207em;"><spanstyle="top:−2.4559em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2441em;"><span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span></span></span></span></span></p><h2><strong>Q:Whatisthefinalanswertotheproblem?</strong></h2><p>A:Thefinalanswertotheproblemis:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>g</mi><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo><mo>=</mo><mn>27</mn><mo>+</mo><mfrac><mn>9</mn><mn>2</mn></mfrac><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">g(x)=27+92log⁡2(x)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">g</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">27</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:2.0074em;vertical−align:−0.686em;"></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3214em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord">2</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord">9</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.686em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop"><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.207em;"><spanstyle="top:−2.4559em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2441em;"><span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span></span></span></span></span></p><h2><strong>Q:Whataresomecommonmistakestoavoidwhenworkingwithlogarithmicfunctions?</strong></h2><p>A:Somecommonmistakestoavoidwhenworkingwithlogarithmicfunctionsinclude:</p><ul><li><strong>Notusingthecorrectbase</strong>:Makesuretousethecorrectbasewhenworkingwithlogarithmicfunctions.</li><li><strong>Notusingthecorrectproperty</strong>:Makesuretousethecorrectpropertyoflogarithmswhenworkingwithlogarithmicfunctions.</li><li><strong>Notsimplifyingtheexpression</strong>:Makesuretosimplifytheexpressionasmuchaspossiblewhenworkingwithlogarithmicfunctions.</li></ul><h2><strong>Q:Whataresomereal−worldapplicationsoflogarithmicfunctions?</strong></h2><p>A:Somereal−worldapplicationsoflogarithmicfunctionsinclude:</p><ul><li><strong>Finance</strong>:Logarithmicfunctionsareusedinfinancetocalculateinterestratesandinvestmentreturns.</li><li><strong>Science</strong>:LogarithmicfunctionsareusedinsciencetocalculatethepHofasolutionandtheconcentrationofasubstance.</li><li><strong>Engineering</strong>:Logarithmicfunctionsareusedinengineeringtocalculatethepowerofasignalandthefrequencyofawave.</li></ul><h2><strong>Conclusion</strong></h2><p>Inconclusion,logarithmicfunctionsareafundamentalconceptinmathematics,andunderstandingtheirpropertiesiscrucialforsolvingvariousmathematicalproblems.Byusingthepropertiesoflogarithms,wecanrewritethelogarithmicfunction<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo><mo>=</mo><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>4</mn></msub><mostretchy="false">(</mo><mn>64</mn><msup><mi>x</mi><mn>9</mn></msup><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">g(x)=log⁡4(64x9)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">g</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.0641em;vertical−align:−0.25em;"></span><spanclass="mop"><spanclass="mop">lo<spanstyle="margin−right:0.01389em;">g</span></span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.207em;"><spanstyle="top:−2.4559em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">4</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2441em;"><span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mord">64</span><spanclass="mord"><spanclass="mordmathnormal">x</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8141em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">9</span></span></span></span></span></span></span></span><spanclass="mclose">)</span></span></span></span>inasimplerform.Wecanalsouselogarithmicfunctionstosolvereal−worldproblemsinfinance,science,andengineering.</p>g(x)=\log _4 (64) + \log _4 (x^9) </span></p> <p>Next, we can use the power property to rewrite the second term:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>4</mn></msub><mo stretchy="false">(</mo><mn>64</mn><mo stretchy="false">)</mo><mo>+</mo><mn>9</mn><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>4</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(x)=\log _4 (64) + 9 \log _4 (x) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.207em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">64</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">9</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.207em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p> <p>Now, we can use the fact that <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mi>b</mi></msub><msup><mi>a</mi><mi>c</mi></msup><mo>=</mo><mi>c</mi><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mi>b</mi></msub><mi>a</mi></mrow><annotation encoding="application/x-tex">\log _b a^c = c \log _b a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9386em;vertical-align:-0.2441em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.242em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9386em;vertical-align:-0.2441em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.242em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">a</span></span></span></span> to rewrite the first term:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>9</mn><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>4</mn></msub><mo stretchy="false">(</mo><msup><mn>2</mn><mn>6</mn></msup><mo stretchy="false">)</mo><mo>+</mo><mn>9</mn><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>4</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(x)=9 \log _4 (2^6) + 9 \log _4 (x) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord">9</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.207em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">9</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.207em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p> <p>Using the property <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mi>b</mi></msub><msup><mi>a</mi><mi>c</mi></msup><mo>=</mo><mi>c</mi><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mi>b</mi></msub><mi>a</mi></mrow><annotation encoding="application/x-tex">\log _b a^c = c \log _b a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9386em;vertical-align:-0.2441em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.242em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">c</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9386em;vertical-align:-0.2441em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.242em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">a</span></span></span></span>, we can rewrite the first term as:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>9</mn><mo>⋅</mo><mn>6</mn><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>4</mn></msub><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>+</mo><mn>9</mn><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>4</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(x)=9 \cdot 6 \log _4 (2) + 9 \log _4 (x) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">9</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">6</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.207em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">2</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">9</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.207em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p> <p>Simplifying the expression, we get:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>54</mn><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>4</mn></msub><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>+</mo><mn>9</mn><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>4</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(x)=54 \log _4 (2) + 9 \log _4 (x) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">54</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.207em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">2</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">9</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.207em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p> <p>However, we can simplify this expression further by using the fact that <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mi>b</mi></msub><mi>a</mi><mo>=</mo><mfrac><mrow><mi>log</mi><mo>⁡</mo><mi>a</mi></mrow><mrow><mi>log</mi><mo>⁡</mo><mi>b</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\log _b a = \frac{\log a}{\log b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9386em;vertical-align:-0.2441em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.242em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.4133em;vertical-align:-0.4811em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">l</span><span class="mtight">o</span><span class="mtight" style="margin-right:0.01389em;">g</span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mathnormal mtight">b</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">l</span><span class="mtight">o</span><span class="mtight" style="margin-right:0.01389em;">g</span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>. This gives us:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>54</mn><mi>log</mi><mo>⁡</mo><mn>2</mn></mrow><mrow><mi>log</mi><mo>⁡</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>9</mn><mi>log</mi><mo>⁡</mo><mi>x</mi></mrow><mrow><mi>log</mi><mo>⁡</mo><mn>4</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">g(x)=\frac{54 \log 2}{\log 4} + \frac{9 \log x}{\log 4} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">54</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p> <p>Using the property <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mo>⁡</mo><msup><mi>a</mi><mi>b</mi></msup><mo>=</mo><mi>b</mi><mi>log</mi><mo>⁡</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">\log a^b = b \log a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0435em;vertical-align:-0.1944em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">a</span></span></span></span>, we can rewrite the expression as:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>54</mn><mi>log</mi><mo>⁡</mo><mn>2</mn></mrow><mrow><mn>2</mn><mi>log</mi><mo>⁡</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>9</mn><mi>log</mi><mo>⁡</mo><mi>x</mi></mrow><mrow><mn>2</mn><mi>log</mi><mo>⁡</mo><mn>2</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">g(x)=\frac{54 \log 2}{2 \log 2} + \frac{9 \log x}{2 \log 2} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">54</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p> <p>Simplifying the expression, we get:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>27</mn><mo>+</mo><mfrac><mn>9</mn><mn>2</mn></mfrac><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(x)=27 + \frac{9}{2} \log _2 (x) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">27</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.207em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p> <p>However, we can simplify this expression further by using the fact that <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mi>b</mi></msub><mi>a</mi><mo>=</mo><mfrac><mrow><mi>log</mi><mo>⁡</mo><mi>a</mi></mrow><mrow><mi>log</mi><mo>⁡</mo><mi>b</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\log _b a = \frac{\log a}{\log b}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9386em;vertical-align:-0.2441em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.242em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.4133em;vertical-align:-0.4811em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">l</span><span class="mtight">o</span><span class="mtight" style="margin-right:0.01389em;">g</span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mathnormal mtight">b</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mtight">l</span><span class="mtight">o</span><span class="mtight" style="margin-right:0.01389em;">g</span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>. This gives us:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>27</mn><mo>+</mo><mfrac><mn>9</mn><mn>2</mn></mfrac><mo>⋅</mo><mfrac><mrow><mi>log</mi><mo>⁡</mo><mi>x</mi></mrow><mrow><mi>log</mi><mo>⁡</mo><mn>2</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">g(x)=27 + \frac{9}{2} \cdot \frac{\log x}{\log 2} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">27</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p> <p>Using the property <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mo>⁡</mo><msup><mi>a</mi><mi>b</mi></msup><mo>=</mo><mi>b</mi><mi>log</mi><mo>⁡</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">\log a^b = b \log a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0435em;vertical-align:-0.1944em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">b</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">a</span></span></span></span>, we can rewrite the expression as:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>27</mn><mo>+</mo><mfrac><mn>9</mn><mn>2</mn></mfrac><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(x)=27 + \frac{9}{2} \log _2 (x) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">27</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.207em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p> <h2><strong>Q: What is the final answer to the problem?</strong></h2> <p>A: The final answer to the problem is:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>27</mn><mo>+</mo><mfrac><mn>9</mn><mn>2</mn></mfrac><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(x)=27 + \frac{9}{2} \log _2 (x) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">27</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.207em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p> <h2><strong>Q: What are some common mistakes to avoid when working with logarithmic functions?</strong></h2> <p>A: Some common mistakes to avoid when working with logarithmic functions include:</p> <ul> <li><strong>Not using the correct base</strong>: Make sure to use the correct base when working with logarithmic functions.</li> <li><strong>Not using the correct property</strong>: Make sure to use the correct property of logarithms when working with logarithmic functions.</li> <li><strong>Not simplifying the expression</strong>: Make sure to simplify the expression as much as possible when working with logarithmic functions.</li> </ul> <h2><strong>Q: What are some real-world applications of logarithmic functions?</strong></h2> <p>A: Some real-world applications of logarithmic functions include:</p> <ul> <li><strong>Finance</strong>: Logarithmic functions are used in finance to calculate interest rates and investment returns.</li> <li><strong>Science</strong>: Logarithmic functions are used in science to calculate the pH of a solution and the concentration of a substance.</li> <li><strong>Engineering</strong>: Logarithmic functions are used in engineering to calculate the power of a signal and the frequency of a wave.</li> </ul> <h2><strong>Conclusion</strong></h2> <p>In conclusion, logarithmic functions are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. By using the properties of logarithms, we can rewrite the logarithmic function <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>4</mn></msub><mo stretchy="false">(</mo><mn>64</mn><msup><mi>x</mi><mn>9</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(x)=\log _4 (64 x^9)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.207em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">64</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> in a simpler form. We can also use logarithmic functions to solve real-world problems in finance, science, and engineering.</p>