Which Expression Is Equivalent To $\log _5\left(\frac{x}{4}\right)^2$?A. $2 \log _5 X + \log _5 4$B. $2 \log _5 X + \log _5 16$C. $2 \log _5 X - 2 \log _5 4$D. $2 \log _5 X - \log _5 4$

Introduction

Logarithmic expressions can be complex and challenging to simplify, but with the right techniques and understanding, they can be broken down into more manageable parts. In this article, we will explore how to simplify the expression $\log _5\left(\frac{x}{4}\right)^2$ and determine which of the given options is equivalent to it.

Understanding Logarithmic Properties

Before we dive into simplifying the expression, it's essential to understand the properties of logarithms. The logarithm of a number to a certain base is the exponent to which the base must be raised to produce that number. In other words, if $y = \log _a x$, then $a^y = x$. This property is the foundation of logarithmic expressions and will be used extensively in this article.

Simplifying the Expression

The given expression is $\log _5\left(\frac{x}{4}\right)^2$. To simplify this expression, we can use the property of logarithms that states $\log _a (b^c) = c \log _a b$. Applying this property to the given expression, we get:

$$\log _5\left(\frac{x}{4}\right)^2 = 2 \log _5\left(\frac{x}{4}\right)$$

Now, we can use the property of logarithms that states $\log _a \left(\frac{b}{c}\right) = \log _a b - \log _a c$. Applying this property to the expression, we get:

$$2 \log _5\left(\frac{x}{4}\right) = 2 \log _5 x - 2 \log _5 4$$

Evaluating the Options

Now that we have simplified the expression, we can evaluate the given options to determine which one is equivalent to it.

• Option A: $2 \log _5 x + \log _5 4$
• Option B: $2 \log _5 x + \log _5 16$
• Option C: $2 \log _5 x - 2 \log _5 4$
• Option D: $2 \log _5 x - \log _5 4$

Comparing the simplified expression with the options, we can see that option C is the only one that matches.

Conclusion

In conclusion, the expression $\log _5\left(\frac{x}{4}\right)^2$ is equivalent to $2 \log _5 x - 2 \log _5 4$. This can be verified by applying the properties of logarithms and simplifying the expression step by step. Understanding logarithmic properties and how to apply them is essential in simplifying complex expressions and solving mathematical problems.

Final Answer

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

A: A logarithmic expression is the inverse of an exponential expression. In other words, if $y = \log _a x$, then $a^y = x$. This means that logarithmic expressions and exponential expressions are related but distinct concepts.

Q: How do I simplify a logarithmic expression with a fraction?

A: To simplify a logarithmic expression with a fraction, you can use the property of logarithms that states $\log _a \left(\frac{b}{c}\right) = \log _a b - \log _a c$. This means that you can break down the fraction into two separate logarithmic expressions and then simplify.

Q: What is the property of logarithms that states $\log _a (b^c) = c \log _a b$?

A: This property is known as the power rule of logarithms. It states that the logarithm of a number raised to a power is equal to the exponent multiplied by the logarithm of the number. In other words, if $y = \log _a (b^c)$, then $y = c \log _a b$.

Q: How do I simplify a logarithmic expression with a product?

A: To simplify a logarithmic expression with a product, you can use the property of logarithms that states $\log _a (b \cdot c) = \log _a b + \log _a c$. This means that you can break down the product into two separate logarithmic expressions and then simplify.

Q: What is the property of logarithms that states $\log _a \left(\frac{b}{c}\right) = \log _a b - \log _a c$?

A: This property is known as the quotient rule of logarithms. It states that the logarithm of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator. In other words, if $y = \log _a \left(\frac{b}{c}\right)$, then $y = \log _a b - \log _a c$.

Q: How do I evaluate a logarithmic expression with a base that is not 10?

A: To evaluate a logarithmic expression with a base that is not 10, you can use the change of base formula. This formula states that $\log _a b = \frac{\log _c b}{\log _c a}$, where $c$ is any positive number not equal to 1. This means that you can change the base of the logarithm to a more familiar base, such as 10, and then evaluate the expression.

Q: What is the change of base formula?

A: The change of base formula is $\log _a b = \frac{\log _c b}{\log _c a}$, where $c$ is any positive number not equal to 1. This formula allows you to change the base of a logarithm to a more familiar base, such as 10, and then evaluate the expression.

Q: How do I use the change of base formula to evaluate a logarithmic expression?

A: To use the change of base formula to evaluate a logarithmic expression, you can follow these steps:

1. Identify the base of the logarithm and the value of the expression.
2. Choose a new base, such as 10, that is more familiar.
3. Use the change of base formula to rewrite the logarithmic expression in terms of the new base.
4. Evaluate the expression using the new base.

Q: What are some common logarithmic expressions that can be simplified using the properties of logarithms?

A: Some common logarithmic expressions that can be simplified using the properties of logarithms include:

• $\log _a (b^c) = c \log _a b$
• $\log _a (b \cdot c) = \log _a b + \log _a c$
• $\log _a \left(\frac{b}{c}\right) = \log _a b - \log _a c$

These properties can be used to simplify a wide range of logarithmic expressions and make them easier to evaluate.

Q: How do I determine which property of logarithms to use when simplifying an expression?

A: To determine which property of logarithms to use when simplifying an expression, you can follow these steps:

1. Identify the type of expression you are working with, such as a product, quotient, or power.
2. Choose the property of logarithms that corresponds to the type of expression.
3. Apply the property to simplify the expression.

By following these steps, you can use the properties of logarithms to simplify a wide range of expressions and make them easier to evaluate.