Which Expression Is Equivalent To $\log _8 \left(4a\frac{b-4}{c^4}\right)?$A. $\log _8 4 + \log _8 A - \log _8(b-4) - 4 \log _8 C$ B. $\log _8 4 + \log _8 A + \left(\log _8(b-4) - 4 \log _8 C\right$\] C. $\log _8 4a +

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 process of simplifying logarithmic expressions, focusing on the given expression $\log _8 \left(4a\frac{b-4}{c^4}\right)$.

Understanding Logarithmic Properties

Before diving into the simplification process, 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 the foundation of our simplification process.

Simplifying the Given Expression

Let's start by applying the properties of logarithms to the given expression $\log _8 \left(4a\frac{b-4}{c^4}\right)$.

Step 1: Apply the Product Property

The given expression can be rewritten as the product of two terms: $4a$ and $\frac{b-4}{c^4}$. Using the product property, we can break down the expression into two separate logarithmic expressions:

$$\log _8 \left(4a\frac{b-4}{c^4}\right) = \log _8 (4a) + \log _8 \left(\frac{b-4}{c^4}\right)$$

Step 2: Apply the Quotient Property

Now, let's focus on the second term, $\log _8 \left(\frac{b-4}{c^4}\right)$. We can apply the quotient property to break it down into two separate logarithmic expressions:

$$\log _8 \left(\frac{b-4}{c^4}\right) = \log _8 (b-4) - \log _8 (c^4)$$

Step 3: Apply the Power Property

The last step is to apply the power property to the second term, $\log _8 (c^4)$. We can rewrite it as:

$$\log _8 (c^4) = 4 \log _8 c$$

Combining the Results

Now that we have broken down the expression into manageable parts, we can combine the results:

$$\log _8 \left(4a\frac{b-4}{c^4}\right) = \log _8 (4a) + \log _8 (b-4) - \log _8 (c^4)$$

$$\log _8 \left(4a\frac{b-4}{c^4}\right) = \log _8 (4a) + \log _8 (b-4) - 4 \log _8 c$$

Conclusion

In this article, we have explored the process of simplifying logarithmic expressions, focusing on the given expression $\log _8 \left(4a\frac{b-4}{c^4}\right)$. By applying the properties of logarithms, we were able to break down the expression into manageable parts and simplify it. The final result is:

$$\log _8 \left(4a\frac{b-4}{c^4}\right) = \log _8 (4a) + \log _8 (b-4) - 4 \log _8 c$$

This expression is equivalent to option A.

Discussion

• Which option is correct? Based on our simplification process, we can conclude that option A is the correct answer.
• Why is option B incorrect? Option B is incorrect because it incorrectly applies the quotient property. The correct application of the quotient property results in $\log _8 (b-4) - \log _8 (c^4)$, not $\log _8 (b-4) + \log _8 (c^4)$.
• Why is option C incorrect? Option C is incorrect because it incorrectly applies the power property. The correct application of the power property results in $4 \log _8 c$, not $\log _8 (c^4)$.

Final Thoughts

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Q: What are the three main properties of logarithms?

A: 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: $\log_b x^y = y \log_b x$

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

A: To apply the product property, you can break down the expression into two separate logarithmic expressions. For example, if you have the expression $\log_b (xy)$, you can rewrite it as $\log_b x + \log_b y$.

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

A: To apply the quotient property, you can break down the expression into two separate logarithmic expressions. For example, if you have the expression $\log_b \left(\frac{x}{y}\right)$, you can rewrite it as $\log_b x - \log_b y$.

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

A: To apply the power property, you can rewrite the expression as the product of the base and the exponent. For example, if you have the expression $\log_b x^y$, you can rewrite it as $y \log_b x$.

Q: What is the difference between a logarithmic expression and an exponential expression?

A: A logarithmic expression is an expression that involves the logarithm of a number, while an exponential expression is an expression that involves the exponentiation of a number. For example, the expression $\log_b x$ is a logarithmic expression, while the expression $b^x$ is an exponential expression.

Q: How do I simplify a logarithmic expression with multiple terms?

A: To simplify a logarithmic expression with multiple terms, you can apply the properties of logarithms to break down the expression into manageable parts. For example, if you have the expression $\log_b (x + y)$, you can rewrite it as $\log_b x + \log_b y$.

Q: What is the final result of the given expression $\log _8 \left(4a\frac{b-4}{c^4}\right)$?

A: The final result of the given expression $\log _8 \left(4a\frac{b-4}{c^4}\right)$ is:

$$\log _8 \left(4a\frac{b-4}{c^4}\right) = \log _8 (4a) + \log _8 (b-4) - 4 \log _8 c$$

Q: Which option is correct?

A: Based on our simplification process, we can conclude that option A is the correct answer.

Q: Why is option B incorrect?

A: Option B is incorrect because it incorrectly applies the quotient property. The correct application of the quotient property results in $\log _8 (b-4) - \log _8 (c^4)$, not $\log _8 (b-4) + \log _8 (c^4)$.

Q: Why is option C incorrect?

A: Option C is incorrect because it incorrectly applies the power property. The correct application of the power property results in $4 \log _8 c$, not $\log _8 (c^4)$.

Conclusion

In this article, we have provided a comprehensive guide to simplifying logarithmic expressions. We have covered the three main properties of logarithms, how to apply them to simplify expressions, and provided a step-by-step guide to simplifying the given expression $\log _8 \left(4a\frac{b-4}{c^4}\right)$. We hope that this article has provided valuable insights and helped you to better understand the process of simplifying logarithmic expressions.