Which Is Equivalent To $3 \log _2 8+4 \log _2 \frac{1}{2}-\log _3 2 ?$A. $5-\log _3 2$B. \$16-\log _3 2$[/tex\]C. $\log _2 48-\log _3 2$D. $\log _3 32$

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Introduction

Logarithmic expressions can be complex and challenging to simplify. However, with a clear understanding of the properties of logarithms, we can break down these expressions into manageable parts. In this article, we will explore the properties of logarithms and apply them to simplify a given expression.

Understanding Logarithmic Properties

Before we dive into the simplification process, let's review the key properties of logarithms:

  • 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
  • Change of Base Property: log⁡bx=log⁡axlog⁡ab\log_b x = \frac{\log_a x}{\log_a b}

Simplifying the Given Expression

The given expression is:

3log⁡28+4log⁡212−log⁡323 \log_2 8 + 4 \log_2 \frac{1}{2} - \log_3 2

To simplify this expression, we will apply the properties of logarithms.

Step 1: Simplify the First Term

Using the power property, we can rewrite the first term as:

3log⁡28=log⁡2833 \log_2 8 = \log_2 8^3

Since 83=5128^3 = 512, we can rewrite the first term as:

log⁡2512\log_2 512

Step 2: Simplify the Second Term

Using the quotient property, we can rewrite the second term as:

4log⁡212=log⁡2(12)44 \log_2 \frac{1}{2} = \log_2 \left(\frac{1}{2}\right)^4

Since (12)4=116\left(\frac{1}{2}\right)^4 = \frac{1}{16}, we can rewrite the second term as:

log⁡2116\log_2 \frac{1}{16}

Step 3: Simplify the Third Term

The third term is already simplified.

Step 4: Combine the Terms

Now, we can combine the simplified terms:

log⁡2512+log⁡2116−log⁡32\log_2 512 + \log_2 \frac{1}{16} - \log_3 2

Using the product property, we can rewrite the first two terms as:

log⁡2(512⋅116)−log⁡32\log_2 (512 \cdot \frac{1}{16}) - \log_3 2

Since 512⋅116=32512 \cdot \frac{1}{16} = 32, we can rewrite the expression as:

log⁡232−log⁡32\log_2 32 - \log_3 2

Step 5: Simplify the Expression

Using the change of base property, we can rewrite the expression as:

log⁡32log⁡2−log⁡2log⁡3\frac{\log 32}{\log 2} - \frac{\log 2}{\log 3}

Since log⁡32=log⁡(25)=5log⁡2\log 32 = \log (2^5) = 5 \log 2, we can rewrite the expression as:

5log⁡2log⁡2−log⁡2log⁡3\frac{5 \log 2}{\log 2} - \frac{\log 2}{\log 3}

Simplifying further, we get:

5−log⁡2log⁡35 - \frac{\log 2}{\log 3}

Step 6: Simplify the Final Expression

Using the change of base property, we can rewrite the expression as:

5−log⁡325 - \log_3 2

Conclusion

In this article, we simplified a given logarithmic expression using the properties of logarithms. We applied the product property, quotient property, power property, and change of base property to break down the expression into manageable parts. The final simplified expression is:

5−log⁡325 - \log_3 2

This expression is equivalent to one of the given options. Let's compare it with the options:

  • A. 5−log⁡325 - \log_3 2
  • B. 16−log⁡3216 - \log_3 2
  • C. log⁡248−log⁡32\log_2 48 - \log_3 2
  • D. log⁡332\log_3 32

Introduction

Logarithmic expressions can be complex and challenging to simplify. However, with a clear understanding of the properties of logarithms, we can break down these expressions into manageable parts. In this article, we will explore the properties of logarithms and apply them to simplify a given expression.

Q&A: Logarithmic Expressions

Q: What is the product property of logarithms?

A: The product property of logarithms states that log⁡b(xy)=log⁡bx+log⁡by\log_b (xy) = \log_b x + \log_b y. This means that the logarithm of a product is equal to the sum of the logarithms of the individual terms.

Q: How do I simplify a logarithmic expression using the product property?

A: To simplify a logarithmic expression using the product property, you can break down the expression into individual terms and then apply the product property. For example, if you have the expression log⁡2(4⋅9)\log_2 (4 \cdot 9), you can break it down into log⁡24+log⁡29\log_2 4 + \log_2 9.

Q: What is the quotient property of logarithms?

A: The quotient property of logarithms states that log⁡b(xy)=log⁡bx−log⁡by\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y. This means that the logarithm of a quotient is equal to the difference of the logarithms of the individual terms.

Q: How do I simplify a logarithmic expression using the quotient property?

A: To simplify a logarithmic expression using the quotient property, you can break down the expression into individual terms and then apply the quotient property. For example, if you have the expression log⁡2(82)\log_2 \left(\frac{8}{2}\right), you can break it down into log⁡28−log⁡22\log_2 8 - \log_2 2.

Q: What is the power property of logarithms?

A: The power property of logarithms states that log⁡bxy=ylog⁡bx\log_b x^y = y \log_b x. This means that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base.

Q: How do I simplify a logarithmic expression using the power property?

A: To simplify a logarithmic expression using the power property, you can break down the expression into individual terms and then apply the power property. For example, if you have the expression log⁡283\log_2 8^3, you can break it down into 3log⁡283 \log_2 8.

Q: What is the change of base property of logarithms?

A: The change of base property of logarithms states that log⁡bx=log⁡axlog⁡ab\log_b x = \frac{\log_a x}{\log_a b}. This means that you can change the base of a logarithm by dividing the logarithm of the number by the logarithm of the new base.

Q: How do I simplify a logarithmic expression using the change of base property?

A: To simplify a logarithmic expression using the change of base property, you can change the base of the logarithm to a more convenient base. For example, if you have the expression log⁡32\log_3 2, you can change the base to base 2 by dividing by log⁡23\log_2 3.

Common Logarithmic Expressions

Q: What is the logarithm of 10?

A: The logarithm of 10 is 1, since 101=1010^1 = 10.

Q: What is the logarithm of 100?

A: The logarithm of 100 is 2, since 102=10010^2 = 100.

Q: What is the logarithm of 1000?

A: The logarithm of 1000 is 3, since 103=100010^3 = 1000.

Conclusion

In this article, we explored the properties of logarithms and applied them to simplify a given expression. We also answered common questions about logarithmic expressions and provided examples of how to simplify them using the product property, quotient property, power property, and change of base property.