Simplify The Expression: $\ln \frac{4y^5}{x^2}$A. $\ln 4 - 2\ln X - 5\ln Y$B. $\ln 4 - 2\ln X + 5\ln Y$C. $-8\ln X + 5\ln Y$

Introduction

In this article, we will simplify the given expression $\ln \frac{4y^5}{x^2}$. This involves applying the properties of logarithms to rewrite the expression in a simpler form. We will use the properties of logarithms, such as the product rule and the quotient rule, to simplify the expression.

Understanding the Properties of Logarithms

Before we simplify the expression, let's review the properties of logarithms that we will use. The product rule states that $\log (ab) = \log a + \log b$, and the quotient rule states that $\log \frac{a}{b} = \log a - \log b$. We will use these properties to simplify the expression.

Simplifying the Expression

To simplify the expression $\ln \frac{4y^5}{x^2}$, we will use the quotient rule. The quotient rule states that $\log \frac{a}{b} = \log a - \log b$. We can rewrite the expression as $\ln 4y^5 - \ln x^2$.

Applying the Product Rule

Now, we will apply the product rule to simplify the expression further. The product rule states that $\log (ab) = \log a + \log b$. We can rewrite the expression as $\ln 4 + \ln y^5 - \ln x^2$.

Simplifying the Expression Further

We can simplify the expression further by using the property of logarithms that states $\log a^b = b\log a$. We can rewrite the expression as $\ln 4 + 5\ln y - 2\ln x$.

Conclusion

In conclusion, we have simplified the expression $\ln \frac{4y^5}{x^2}$ using the properties of logarithms. We used the quotient rule and the product rule to rewrite the expression in a simpler form. The final simplified expression is $\ln 4 + 5\ln y - 2\ln x$.

Answer

The correct answer is $\ln 4 + 5\ln y - 2\ln x$. This is the simplified form of the expression $\ln \frac{4y^5}{x^2}$.

Comparison with Other Options

Let's compare our answer with the other options.

• Option A: $\ln 4 - 2\ln x - 5\ln y$
• Option B: $\ln 4 - 2\ln x + 5\ln y$
• Option C: $-8\ln x + 5\ln y$

Our answer is different from all the other options. This is because we used the correct properties of logarithms to simplify the expression.

Conclusion

In conclusion, we have simplified the expression $\ln \frac{4y^5}{x^2}$ using the properties of logarithms. We used the quotient rule and the product rule to rewrite the expression in a simpler form. The final simplified expression is $\ln 4 + 5\ln y - 2\ln x$. This is the correct answer.

Final Answer

Introduction

In our previous article, we simplified the expression $\ln \frac{4y^5}{x^2}$ using the properties of logarithms. We used the quotient rule and the product rule to rewrite the expression in a simpler form. In this article, we will answer some common questions related to the simplification of the expression.

Q&A

Q: What is the quotient rule of logarithms?

A: The quotient rule of logarithms states that $\log \frac{a}{b} = \log a - \log b$. This means that the logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor.

Q: What is the product rule of logarithms?

A: The product rule of logarithms states that $\log (ab) = \log a + \log b$. This means that the logarithm of a product is equal to the logarithm of the first factor plus the logarithm of the second factor.

Q: How do I apply the quotient rule to simplify the expression $\ln \frac{4y^5}{x^2}$?

A: To apply the quotient rule, you need to rewrite the expression as $\ln 4y^5 - \ln x^2$. Then, you can use the product rule to simplify the expression further.

Q: What is the final simplified expression of $\ln \frac{4y^5}{x^2}$?

A: The final simplified expression of $\ln \frac{4y^5}{x^2}$ is $\ln 4 + 5\ln y - 2\ln x$.

Q: Why is the final simplified expression different from the other options?

A: The final simplified expression is different from the other options because we used the correct properties of logarithms to simplify the expression. The other options do not use the correct properties of logarithms, so they are incorrect.

Q: Can I use the quotient rule and the product rule to simplify any expression?

A: Yes, you can use the quotient rule and the product rule to simplify any expression that involves logarithms. However, you need to make sure that you apply the rules correctly and use the correct properties of logarithms.

Q: What are some common mistakes to avoid when simplifying expressions involving logarithms?

A: Some common mistakes to avoid when simplifying expressions involving logarithms include:

• Not using the correct properties of logarithms
• Not applying the quotient rule and the product rule correctly
• Not simplifying the expression further using the properties of logarithms

Conclusion

In conclusion, we have answered some common questions related to the simplification of the expression $\ln \frac{4y^5}{x^2}$. We used the quotient rule and the product rule to rewrite the expression in a simpler form. The final simplified expression is $\ln 4 + 5\ln y - 2\ln x$. We also discussed some common mistakes to avoid when simplifying expressions involving logarithms.

Final Answer

The final answer is $\boxed{\ln 4 + 5\ln y - 2\ln x}$.