Select The Correct Answer.Which Expression Is Equivalent To The Given Expression? $\ln \left(\frac{2 E}{x}\right$\]A. $\ln 2 - \ln X$ B. $\ln 2 + \ln X$ C. $\ln 1 + \ln 2 - \ln X$ D. $1 + \ln 2 - \ln X$

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**Select the Correct Answer: Which Expression is Equivalent to the Given Expression?**

In mathematics, logarithmic expressions are a crucial part of various mathematical operations. Understanding how to manipulate and simplify these expressions is essential for solving complex problems. In this article, we will focus on finding the equivalent expression for a given logarithmic expression.

The Given Expression

The given expression is ln⁡(2ex)\ln \left(\frac{2 e}{x}\right). Our goal is to find an equivalent expression for this given expression.

Understanding the Properties of Logarithms

Before we proceed, let's recall some essential properties of logarithms:

  • Product Rule: log⁡b(m⋅n)=log⁡bm+log⁡bn\log_b (m \cdot n) = \log_b m + \log_b n
  • Quotient Rule: log⁡b(mn)=log⁡bm−log⁡bn\log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n
  • Power Rule: log⁡b(mp)=p⋅log⁡bm\log_b (m^p) = p \cdot \log_b m

Step 1: Apply the Quotient Rule

Using the Quotient Rule, we can rewrite the given expression as:

ln⁡(2ex)=ln⁡2e−ln⁡x\ln \left(\frac{2 e}{x}\right) = \ln 2 e - \ln x

Step 2: Apply the Product Rule

Now, let's apply the Product Rule to the first term:

ln⁡2e=ln⁡2+ln⁡e\ln 2 e = \ln 2 + \ln e

Step 3: Simplify the Expression

Since ln⁡e=1\ln e = 1, we can simplify the expression as:

ln⁡2+ln⁡e=ln⁡2+1\ln 2 + \ln e = \ln 2 + 1

Step 4: Combine the Terms

Now, let's combine the terms:

ln⁡2+1−ln⁡x=1+ln⁡2−ln⁡x\ln 2 + 1 - \ln x = 1 + \ln 2 - \ln x

Therefore, the equivalent expression for the given expression ln⁡(2ex)\ln \left(\frac{2 e}{x}\right) is:

1+ln⁡2−ln⁡x1 + \ln 2 - \ln x

Q: What is the Quotient Rule for logarithms?

A: The Quotient Rule for logarithms states that log⁡b(mn)=log⁡bm−log⁡bn\log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n.

Q: How do you apply the Product Rule for logarithms?

A: To apply the Product Rule, you need to multiply the logarithms of the two terms. For example, log⁡b(m⋅n)=log⁡bm+log⁡bn\log_b (m \cdot n) = \log_b m + \log_b n.

Q: What is the Power Rule for logarithms?

A: The Power Rule for logarithms states that log⁡b(mp)=p⋅log⁡bm\log_b (m^p) = p \cdot \log_b m.

Q: Can you provide an example of how to simplify a logarithmic expression using the Quotient Rule?

A: Yes, here's an example:

ln⁡(34)=ln⁡3−ln⁡4\ln \left(\frac{3}{4}\right) = \ln 3 - \ln 4

Q: How do you simplify a logarithmic expression using the Product Rule?

A: Here's an example:

ln⁡(2⋅3)=ln⁡2+ln⁡3\ln (2 \cdot 3) = \ln 2 + \ln 3

Q: What is the final answer to the given expression?

A: The final answer is 1+ln⁡2−ln⁡x1 + \ln 2 - \ln x.

Q: What is the difference between the Quotient Rule and the Product Rule for logarithms?

A: The Quotient Rule is used to simplify expressions with fractions, while the Product Rule is used to simplify expressions with products.

Q: Can you provide more examples of how to simplify logarithmic expressions using the Quotient Rule and the Product Rule?

A: Yes, here are some more examples:

  • ln⁡(23)=ln⁡2−ln⁡3\ln \left(\frac{2}{3}\right) = \ln 2 - \ln 3
  • ln⁡(2⋅3)=ln⁡2+ln⁡3\ln (2 \cdot 3) = \ln 2 + \ln 3
  • ln⁡(45)=ln⁡4−ln⁡5\ln \left(\frac{4}{5}\right) = \ln 4 - \ln 5

In conclusion, understanding the properties of logarithms, such as the Quotient Rule and the Product Rule, is essential for simplifying logarithmic expressions. By applying these rules, you can simplify complex expressions and arrive at the correct answer.