Expand Each Expression.1. $\ln (2x)^4$A. $4 \ln 2 + 4 \ln X$ B. $4 \ln 2 + \ln X$ C. $8 \ln X$

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Introduction

Logarithmic expressions are a fundamental concept in mathematics, and expanding them is a crucial skill to master. In this article, we will focus on expanding the expression ln⁡(2x)4\ln (2x)^4 and explore the different possible solutions. We will also discuss the properties of logarithms and how they can be used to simplify complex expressions.

Properties of Logarithms

Before we dive into expanding the expression, let's review the properties of logarithms. The logarithm of a product can be expressed as the sum of the logarithms of the individual terms. This property is known as the product rule:

log⁡(ab)=log⁡a+log⁡b\log (ab) = \log a + \log b

The logarithm of a quotient can be expressed as the difference of the logarithms of the individual terms. This property is known as the quotient rule:

log⁡(ab)=log⁡a−log⁡b\log \left(\frac{a}{b}\right) = \log a - \log b

The logarithm of a power can be expressed as the exponent multiplied by the logarithm of the base. This property is known as the power rule:

log⁡ab=blog⁡a\log a^b = b \log a

Expanding the Expression

Now that we have reviewed the properties of logarithms, let's focus on expanding the expression ln⁡(2x)4\ln (2x)^4. Using the power rule, we can rewrite the expression as:

ln⁡(2x)4=4ln⁡(2x)\ln (2x)^4 = 4 \ln (2x)

Using the product rule, we can rewrite the expression as:

4ln⁡(2x)=4(ln⁡2+ln⁡x)4 \ln (2x) = 4 (\ln 2 + \ln x)

Using the power rule again, we can rewrite the expression as:

4(ln⁡2+ln⁡x)=4ln⁡2+4ln⁡x4 (\ln 2 + \ln x) = 4 \ln 2 + 4 \ln x

Therefore, the expanded expression is:

ln⁡(2x)4=4ln⁡2+4ln⁡x\ln (2x)^4 = 4 \ln 2 + 4 \ln x

Alternative Solutions

However, there are alternative solutions to this problem. Let's explore them.

Solution A

One possible solution is to use the property of logarithms that states log⁡ab=blog⁡a\log a^b = b \log a. Applying this property to the expression, we get:

ln⁡(2x)4=4ln⁡(2x)\ln (2x)^4 = 4 \ln (2x)

Using the product rule, we can rewrite the expression as:

4ln⁡(2x)=4(ln⁡2+ln⁡x)4 \ln (2x) = 4 (\ln 2 + \ln x)

Using the power rule again, we can rewrite the expression as:

4(ln⁡2+ln⁡x)=4ln⁡2+4ln⁡x4 (\ln 2 + \ln x) = 4 \ln 2 + 4 \ln x

Therefore, the expanded expression is:

ln⁡(2x)4=4ln⁡2+4ln⁡x\ln (2x)^4 = 4 \ln 2 + 4 \ln x

Solution B

Another possible solution is to use the property of logarithms that states log⁡(ab)=log⁡a−log⁡b\log \left(\frac{a}{b}\right) = \log a - \log b. Applying this property to the expression, we get:

ln⁡(2x)4=ln⁡((2x)41)\ln (2x)^4 = \ln \left(\frac{(2x)^4}{1}\right)

Using the quotient rule, we can rewrite the expression as:

ln⁡((2x)41)=ln⁡(2x)4−ln⁡1\ln \left(\frac{(2x)^4}{1}\right) = \ln (2x)^4 - \ln 1

Using the property of logarithms that states log⁡1=0\log 1 = 0, we can rewrite the expression as:

ln⁡(2x)4−ln⁡1=ln⁡(2x)4\ln (2x)^4 - \ln 1 = \ln (2x)^4

Using the power rule, we can rewrite the expression as:

ln⁡(2x)4=4ln⁡(2x)\ln (2x)^4 = 4 \ln (2x)

Using the product rule, we can rewrite the expression as:

4ln⁡(2x)=4(ln⁡2+ln⁡x)4 \ln (2x) = 4 (\ln 2 + \ln x)

Using the power rule again, we can rewrite the expression as:

4(ln⁡2+ln⁡x)=4ln⁡2+4ln⁡x4 (\ln 2 + \ln x) = 4 \ln 2 + 4 \ln x

Therefore, the expanded expression is:

ln⁡(2x)4=4ln⁡2+4ln⁡x\ln (2x)^4 = 4 \ln 2 + 4 \ln x

Solution C

Another possible solution is to use the property of logarithms that states log⁡ab=blog⁡a\log a^b = b \log a. Applying this property to the expression, we get:

ln⁡(2x)4=4ln⁡(2x)\ln (2x)^4 = 4 \ln (2x)

Using the product rule, we can rewrite the expression as:

4ln⁡(2x)=4(ln⁡2+ln⁡x)4 \ln (2x) = 4 (\ln 2 + \ln x)

Using the power rule again, we can rewrite the expression as:

4(ln⁡2+ln⁡x)=4ln⁡2+4ln⁡x4 (\ln 2 + \ln x) = 4 \ln 2 + 4 \ln x

However, this solution is incorrect. The correct solution is:

ln⁡(2x)4=4ln⁡2+ln⁡x\ln (2x)^4 = 4 \ln 2 + \ln x

Conclusion

In conclusion, expanding the expression ln⁡(2x)4\ln (2x)^4 requires a deep understanding of the properties of logarithms. By applying the power rule, product rule, and quotient rule, we can simplify complex expressions and arrive at the correct solution. In this article, we have explored three different solutions to this problem, and we have seen that the correct solution is:

ln⁡(2x)4=4ln⁡2+4ln⁡x\ln (2x)^4 = 4 \ln 2 + 4 \ln x

Final Answer

The final answer is:

\boxed{4 \ln 2 + 4 \ln x}$<br/> **Logarithmic Expressions: A Q&A Guide** ===================================== **Introduction** --------------- Logarithmic expressions are a fundamental concept in mathematics, and understanding them is crucial for solving complex problems. In this article, we will provide a Q&A guide to help you better understand logarithmic expressions and how to expand them. **Q: What is a logarithmic expression?** -------------------------------------- A: A logarithmic expression is an expression that involves a logarithm, which is the inverse operation of exponentiation. Logarithmic expressions are used to solve problems that involve exponential growth or decay. **Q: What are the properties of logarithms?** ----------------------------------------- A: The properties of logarithms are: * The product rule: $\log (ab) = \log a + \log b$ * The quotient rule: $\log \left(\frac{a}{b}\right) = \log a - \log b$ * The power rule: $\log a^b = b \log a$ **Q: How do I expand a logarithmic expression?** -------------------------------------------- A: To expand a logarithmic expression, you can use the properties of logarithms. Here are the steps: 1. Identify the logarithmic expression and the properties of logarithms that can be applied. 2. Apply the product rule, quotient rule, or power rule to simplify the expression. 3. Use the properties of logarithms to rewrite the expression in a simpler form. **Q: What is the difference between $\ln (2x)^4$ and $\ln 2 + \ln x + \ln x + \ln x + \ln x + \ln x$?** --------------------------------------------------------- A: The expression $\ln (2x)^4$ is equivalent to $\ln 2 + \ln x + \ln x + \ln x + \ln x + \ln x$, but it is not equal to $4 \ln 2 + 4 \ln x$. The correct expansion of $\ln (2x)^4$ is $4 \ln 2 + 4 \ln x$. **Q: How do I simplify a logarithmic expression with multiple terms?** --------------------------------------------------------- A: To simplify a logarithmic expression with multiple terms, you can use the properties of logarithms. Here are the steps: 1. Identify the logarithmic expression and the properties of logarithms that can be applied. 2. Apply the product rule, quotient rule, or power rule to simplify the expression. 3. Use the properties of logarithms to rewrite the expression in a simpler form. **Q: What is the final answer to the expression $\ln (2x)^4$?** --------------------------------------------------------- A: The final answer to the expression $\ln (2x)^4$ is $4 \ln 2 + 4 \ln x$. **Q: Can I use the power rule to simplify a logarithmic expression with a negative exponent?** --------------------------------------------------------- A: No, you cannot use the power rule to simplify a logarithmic expression with a negative exponent. The power rule only applies to positive exponents. **Q: How do I evaluate a logarithmic expression with a variable base?** --------------------------------------------------------- A: To evaluate a logarithmic expression with a variable base, you can use the properties of logarithms. Here are the steps: 1. Identify the logarithmic expression and the properties of logarithms that can be applied. 2. Apply the product rule, quotient rule, or power rule to simplify the expression. 3. Use the properties of logarithms to rewrite the expression in a simpler form. **Conclusion** ---------- In conclusion, logarithmic expressions are a fundamental concept in mathematics, and understanding them is crucial for solving complex problems. By applying the properties of logarithms and following the steps outlined in this article, you can simplify complex logarithmic expressions and arrive at the correct solution.