Rewrite The Following Expression As A Single Logarithm With A Coefficient Of 1.Given: $3 \log _5 \frac{u V^2}{w^3}$A. $\log _5 U V^2$ B. $\log _5 \frac{u V^6}{w^9}$ C. $\log _5 \frac{u^3 V^6}{w^3}$ D. $\log _5

Introduction

In this article, we will focus on rewriting a given logarithmic expression as a single logarithm with a coefficient of 1. This involves applying the properties of logarithms to simplify the expression and express it in a more compact form. We will use the given expression $3 \log _5 \frac{u v^2}{w^3}$ and rewrite it as a single logarithm with a coefficient of 1.

Understanding the Properties of Logarithms

Before we proceed with rewriting the given expression, it is essential to understand the properties of logarithms. The properties of logarithms are as follows:

• Product Property: $\log _a (m \cdot n) = \log _a m + \log _a n$
• Quotient Property: $\log _a \frac{m}{n} = \log _a m - \log _a n$
• Power Property: $\log _a m^p = p \cdot \log _a m$

These properties will be used to rewrite the given expression as a single logarithm with a coefficient of 1.

Rewriting the Given Expression

The given expression is $3 \log _5 \frac{u v^2}{w^3}$. To rewrite this expression as a single logarithm with a coefficient of 1, we will apply the properties of logarithms.

First, we will use the quotient property to rewrite the expression as follows:

$$3 \log _5 \frac{u v^2}{w^3} = 3 (\log _5 u v^2 - \log _5 w^3)$$

Next, we will use the product property to rewrite the expression as follows:

$$3 (\log _5 u v^2 - \log _5 w^3) = 3 \log _5 u v^2 - 3 \log _5 w^3$$

Now, we will use the power property to rewrite the expression as follows:

$$3 \log _5 u v^2 - 3 \log _5 w^3 = \log _5 u^3 v^6 - \log _5 w^9$$

Finally, we will use the quotient property to rewrite the expression as follows:

$$\log _5 u^3 v^6 - \log _5 w^9 = \log _5 \frac{u^3 v^6}{w^9}$$

Conclusion

In this article, we have rewritten the given logarithmic expression $3 \log _5 \frac{u v^2}{w^3}$ as a single logarithm with a coefficient of 1. We have applied the properties of logarithms to simplify the expression and express it in a more compact form. The rewritten expression is $\log _5 \frac{u^3 v^6}{w^9}$.

Answer

The correct answer is C. $\log _5 \frac{u^3 v^6}{w^9}$.

Discussion

The given expression $3 \log _5 \frac{u v^2}{w^3}$ can be rewritten as a single logarithm with a coefficient of 1 using the properties of logarithms. The rewritten expression is $\log _5 \frac{u^3 v^6}{w^9}$. This expression can be further simplified by applying the properties of logarithms.

Example Use Case

The rewritten expression $\log _5 \frac{u^3 v^6}{w^9}$ can be used in various mathematical applications, such as solving equations and inequalities involving logarithms. For example, if we have the equation $\log _5 \frac{u^3 v^6}{w^9} = 2$, we can solve for $u$, $v$, and $w$ by applying the properties of logarithms.

Step-by-Step Solution

To rewrite the given expression $3 \log _5 \frac{u v^2}{w^3}$ as a single logarithm with a coefficient of 1, follow these steps:

1. Use the quotient property to rewrite the expression as follows: $3 \log _5 \frac{u v^2}{w^3} = 3 (\log _5 u v^2 - \log _5 w^3)$
2. Use the product property to rewrite the expression as follows: $3 (\log _5 u v^2 - \log _5 w^3) = 3 \log _5 u v^2 - 3 \log _5 w^3$
3. Use the power property to rewrite the expression as follows: $3 \log _5 u v^2 - 3 \log _5 w^3 = \log _5 u^3 v^6 - \log _5 w^9$
4. Use the quotient property to rewrite the expression as follows: $\log _5 u^3 v^6 - \log _5 w^9 = \log _5 \frac{u^3 v^6}{w^9}$

Q: What is the product property of logarithms?

A: The product property of logarithms states that $\log _a (m \cdot n) = \log _a m + \log _a n$. This means that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

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

A: To apply the product property, you need to identify the factors in the logarithmic expression and rewrite it as the sum of the logarithms of the individual factors. For example, if you have the expression $\log _5 (u \cdot v^2)$, you can rewrite it as $\log _5 u + \log _5 v^2$.

Q: What is the quotient property of logarithms?

A: The quotient property of logarithms states that $\log _a \frac{m}{n} = \log _a m - \log _a n$. This means that the logarithm of a quotient is equal to the difference of the logarithms of the individual factors.

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

A: To apply the quotient property, you need to identify the factors in the logarithmic expression and rewrite it as the difference of the logarithms of the individual factors. For example, if you have the expression $\log _5 \frac{u}{v^2}$, you can rewrite it as $\log _5 u - \log _5 v^2$.

Q: What is the power property of logarithms?

A: The power property of logarithms states that $\log _a m^p = p \cdot \log _a m$. This means that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base.

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

A: To apply the power property, you need to identify the exponent in the logarithmic expression and rewrite it as the product of the exponent and the logarithm of the base. For example, if you have the expression $\log _5 u^3$, you can rewrite it as $3 \log _5 u$.

Q: How do I rewrite a logarithmic expression with a coefficient?

A: To rewrite a logarithmic expression with a coefficient, you need to apply the properties of logarithms to simplify the expression. For example, if you have the expression $3 \log _5 \frac{u v^2}{w^3}$, you can rewrite it as $\log _5 \frac{u^3 v^6}{w^9}$.

Q: What are some common mistakes to avoid when rewriting logarithmic expressions?

A: Some common mistakes to avoid when rewriting logarithmic expressions include:

• Not applying the properties of logarithms correctly
• Not simplifying the expression enough
• Not checking the domain of the logarithmic function
• Not using the correct base for the logarithm

Q: How do I check my work when rewriting logarithmic expressions?

A: To check your work when rewriting logarithmic expressions, you need to:

• Verify that you have applied the properties of logarithms correctly
• Simplify the expression to ensure that it is in the correct form
• Check the domain of the logarithmic function to ensure that it is valid
• Use a calculator or computer software to verify the result

By following these steps and avoiding common mistakes, you can ensure that your work is accurate and reliable.