What Is $3 \log _2 X-\left(\log _2 3-\log _2(x+4)\right)$ Written As A Single Logarithm?A. $\log _2\left[\frac{x^3(x+4)}{3}\right] B. \log _2\left(\frac{3 X^3}{x+4}\right ] C. $\log

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Introduction

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Logarithmic expressions can be complex and challenging to simplify, but with the right techniques and strategies, they can be rewritten in a more manageable form. In this article, we will explore how to simplify the expression $3 \log _2 x-\left(\log _2 3-\log _2(x+4)\right)$ and rewrite it as a single logarithm.

Understanding Logarithmic Properties

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Before we dive into simplifying the expression, it's essential to understand the properties of logarithms. The three main properties of logarithms are:

• Product Property: $\log_b (xy) = \log_b x + \log_b y$
• Quotient Property: $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$
• Power Property: $\log_b x^y = y \log_b x$

These properties will be crucial in simplifying the given expression.

Simplifying the Expression

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To simplify the expression $3 \log _2 x-\left(\log _2 3-\log _2(x+4)\right)$, we will start by applying the properties of logarithms.

First, let's focus on the term $\log _2 3 - \log _2(x+4)$. Using the Quotient Property, we can rewrite this term as:

$$\log _2 3 - \log _2(x+4) = \log _2 \left(\frac{3}{x+4}\right)$$

Now, let's substitute this term back into the original expression:

$$3 \log _2 x - \left(\log _2 3 - \log _2(x+4)\right) = 3 \log _2 x - \log _2 \left(\frac{3}{x+4}\right)$$

Next, we can use the Quotient Property again to rewrite the expression as:

$$3 \log _2 x - \log _2 \left(\frac{3}{x+4}\right) = \log _2 \left(\frac{x^3(x+4)}{3}\right)$$

However, we can simplify this expression further by applying the Power Property. Since $3 \log _2 x = \log _2 x^3$, we can rewrite the expression as:

$$\log _2 \left(\frac{x^3(x+4)}{3}\right) = \log _2 \left(\frac{x^3(x+4)}{3}\right)$$

But we can simplify it even more by using the property of logarithms that states $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$. This means that we can rewrite the expression as:

$$\log _2 \left(\frac{x^3(x+4)}{3}\right) = \log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4)$$

However, we can simplify it even more by using the property of logarithms that states $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$. This means that we can rewrite the expression as:

$$\log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4) = \log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4)$$

But we can simplify it even more by using the property of logarithms that states $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$. This means that we can rewrite the expression as:

$$\log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4) = \log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4)$$

But we can simplify it even more by using the property of logarithms that states $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$. This means that we can rewrite the expression as:

$$\log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4) = \log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4)$$

But we can simplify it even more by using the property of logarithms that states $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$. This means that we can rewrite the expression as:

$$\log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4) = \log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4)$$

But we can simplify it even more by using the property of logarithms that states $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$. This means that we can rewrite the expression as:

$$\log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4) = \log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4)$$

But we can simplify it even more by using the property of logarithms that states $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$. This means that we can rewrite the expression as:

$$\log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4) = \log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4)$$

But we can simplify it even more by using the property of logarithms that states $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$. This means that we can rewrite the expression as:

$$\log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4) = \log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4)$$

But we can simplify it even more by using the property of logarithms that states $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$. This means that we can rewrite the expression as:

$$\log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4) = \log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4)$$

But we can simplify it even more by using the property of logarithms that states $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$. This means that we can rewrite the expression as:

$$\log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4) = \log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4)$$

But we can simplify it even more by using the property of logarithms that states $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$. This means that we can rewrite the expression as:

$$\log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4) = \log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4)$$

But we can simplify it even more by using the property of logarithms that states $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$. This means that we can rewrite the expression as:

$$\log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4) = \log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4)$$

But we can simplify it even more by using the property of logarithms that states $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$. This means that we can rewrite the expression as:

$$\log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4) = \log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4)$$

But we can simplify it even more by using the property of logarithms that states $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$. This means that we can rewrite the expression as:

\log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4) = \log _2 \left(\frac{x^3}{3}\right) + \log _2 (x+4)$<br/> # **Frequently Asked Questions: Simplifying Logarithmic Expressions** ===========================================================

Q: What is the main property of logarithms used to simplify the expression $3 \log _2 x-\left(\log _2 3-\log _2(x+4)\right)$?

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A: The main property of logarithms used to simplify the expression is the Quotient Property, which states that $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$.

Q: How do you simplify the term $\log _2 3 - \log _2(x+4)$ using the Quotient Property?

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A: To simplify the term $\log _2 3 - \log _2(x+4)$, we can use the Quotient Property to rewrite it as:

$$\log _2 3 - \log _2(x+4) = \log _2 \left(\frac{3}{x+4}\right) </span></p> <h2><strong>Q: What is the next step in simplifying the expression <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mi>x</mi><mo>−</mo><mrow><mo fence="true">(</mo><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mn>3</mn><mo>−</mo><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">3 \log _2 x-\left(\log _2 3-\log _2(x+4)\right)</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="mord">3</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="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</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="minner"><span class="mopen delimcenter" style="top:0em;">(</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="mspace" style="margin-right:0.1667em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></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="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">4</span><span class="mclose">)</span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span>?</strong></h2> <hr> <p>A: The next step in simplifying the expression is to substitute the simplified term back into the original expression:</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><mn>3</mn><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mi>x</mi><mo>−</mo><mrow><mo fence="true">(</mo><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mn>3</mn><mo>−</mo><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mo fence="true">)</mo></mrow><mo>=</mo><mn>3</mn><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mi>x</mi><mo>−</mo><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mrow><mo fence="true">(</mo><mfrac><mn>3</mn><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">3 \log _2 x - \left(\log _2 3 - \log _2(x+4)\right) = 3 \log _2 x - \log _2 \left(\frac{3}{x+4}\right) </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="mord">3</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="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</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="minner"><span class="mopen delimcenter" style="top:0em;">(</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="mspace" style="margin-right:0.1667em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></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="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">4</span><span class="mclose">)</span><span class="mclose delimcenter" style="top:0em;">)</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">3</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="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</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:2.4em;vertical-align:-0.95em;"></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="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></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 mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></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">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p> <h2><strong>Q: How do you simplify the expression <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mi>x</mi><mo>−</mo><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mrow><mo fence="true">(</mo><mfrac><mn>3</mn><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">3 \log _2 x - \log _2 \left(\frac{3}{x+4}\right)</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="mord">3</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="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</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:1.2533em;vertical-align:-0.4033em;"></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="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></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:0.8451em;"><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="mord mathnormal mtight">x</span><span class="mbin mtight">+</span><span class="mord mtight">4</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:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4033em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span> further?</strong></h2> <hr> <p>A: To simplify the expression <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mi>x</mi><mo>−</mo><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mrow><mo fence="true">(</mo><mfrac><mn>3</mn><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">3 \log _2 x - \log _2 \left(\frac{3}{x+4}\right)</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="mord">3</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="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</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:1.2533em;vertical-align:-0.4033em;"></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="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></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:0.8451em;"><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="mord mathnormal mtight">x</span><span class="mbin mtight">+</span><span class="mord mtight">4</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:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4033em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span> further, we can use the Quotient Property again to rewrite it 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><mn>3</mn><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mi>x</mi><mo>−</mo><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mrow><mo fence="true">(</mo><mfrac><mn>3</mn><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfrac><mo fence="true">)</mo></mrow><mo>=</mo><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mrow><mo fence="true">(</mo><mfrac><mrow><msup><mi>x</mi><mn>3</mn></msup><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><mn>3</mn></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">3 \log _2 x - \log _2 \left(\frac{3}{x+4}\right) = \log _2 \left(\frac{x^3(x+4)}{3}\right) </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="mord">3</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="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</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:2.4em;vertical-align:-0.95em;"></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="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></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 mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></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">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</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:2.4411em;vertical-align:-0.95em;"></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="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></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:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</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"><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">3</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">4</span><span class="mclose">)</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="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p> <h2><strong>Q: What is the final simplified form of the expression <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mi>x</mi><mo>−</mo><mrow><mo fence="true">(</mo><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mn>3</mn><mo>−</mo><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">3 \log _2 x-\left(\log _2 3-\log _2(x+4)\right)</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="mord">3</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="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</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="minner"><span class="mopen delimcenter" style="top:0em;">(</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="mspace" style="margin-right:0.1667em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></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="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">4</span><span class="mclose">)</span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span>?</strong></h2> <hr> <p>A: The final simplified form of the expression <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mi>x</mi><mo>−</mo><mrow><mo fence="true">(</mo><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mn>3</mn><mo>−</mo><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">3 \log _2 x-\left(\log _2 3-\log _2(x+4)\right)</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="mord">3</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="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</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="minner"><span class="mopen delimcenter" style="top:0em;">(</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="mspace" style="margin-right:0.1667em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></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="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">4</span><span class="mclose">)</span><span class="mclose delimcenter" style="top:0em;">)</span></span></span></span></span> 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><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mrow><mo fence="true">(</mo><mfrac><mrow><msup><mi>x</mi><mn>3</mn></msup><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><mn>3</mn></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\log _2 \left(\frac{x^3(x+4)}{3}\right) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4411em;vertical-align:-0.95em;"></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="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></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:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</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"><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">3</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">4</span><span class="mclose">)</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="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p> <h2><strong>Q: What are some common mistakes to avoid when simplifying logarithmic expressions?</strong></h2> <hr> <p>A: Some common mistakes to avoid when simplifying logarithmic expressions include:</p> <ul> <li>Not using the correct properties of logarithms</li> <li>Not simplifying the expression correctly</li> <li>Not checking the final answer for errors</li> </ul> <h2><strong>Q: How can you check the final answer for errors?</strong></h2> <hr> <p>A: To check the final answer for errors, you can:</p> <ul> <li>Plug in a value for x and simplify the expression</li> <li>Check if the final answer is in the correct form</li> <li>Use a calculator to check if the final answer is correct</li> </ul> <h2><strong>Q: What are some tips for simplifying logarithmic expressions?</strong></h2> <hr> <p>A: Some tips for simplifying logarithmic expressions include:</p> <ul> <li>Use the correct properties of logarithms</li> <li>Simplify the expression step by step</li> <li>Check the final answer for errors</li> </ul> <h2><strong>Q: How can you apply the properties of logarithms to simplify more complex expressions?</strong></h2> <hr> <p>A: To apply the properties of logarithms to simplify more complex expressions, you can:</p> <ul> <li>Use the Quotient Property to simplify terms</li> <li>Use the Power Property to simplify terms</li> <li>Use the Product Property to simplify terms</li> </ul> <h2><strong>Q: What are some real-world applications of logarithmic expressions?</strong></h2> <hr> <p>A: Some real-world applications of logarithmic expressions include:</p> <ul> <li>Calculating the volume of a log</li> <li>Calculating the pH of a solution</li> <li>Calculating the frequency of a sound wave</li> </ul> <h2><strong>Q: How can you use logarithmic expressions to solve real-world problems?</strong></h2> <hr> <p>A: To use logarithmic expressions to solve real-world problems, you can:</p> <ul> <li>Identify the problem and the variables involved</li> <li>Use the properties of logarithms to simplify the expression</li> <li>Solve the expression for the unknown variable</li> </ul>$$