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

Introduction

Logarithmic expressions can be complex and challenging to simplify. However, with a clear understanding of the properties of logarithms, we can break down these expressions into more manageable parts. In this article, we will explore how to simplify a given logarithmic expression and identify the correct equivalent expression from a list of options.

Understanding Logarithmic Properties

Before we dive into the problem, let's review some essential logarithmic properties:

• Product Rule: $\log_b (xy) = \log_b x + \log_b y$
• Quotient Rule: $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$
• Power Rule: $\log_b (x^y) = y \log_b x$

These properties will be crucial in simplifying the given expression.

The Given Expression

The given expression is:

$\ln \left(\frac{2 e}{x}\right)$

Our goal is to simplify this expression and identify the correct equivalent expression from the list of options.

Step 1: Apply the Quotient Rule

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

$\ln \left(\frac{2 e}{x}\right) = \ln 2 e - \ln x$

Step 2: Apply the Product Rule

Now, we can apply the Product Rule to simplify the expression further:

$\ln 2 e - \ln x = \ln 2 + \ln e - \ln x$

Step 3: Simplify Using the Properties of Logarithms

We know that $\ln e = 1$, so we can substitute this value into the expression:

$\ln 2 + \ln e - \ln x = \ln 2 + 1 - \ln x$

Step 4: Simplify the Expression

Now, we can simplify the expression by combining the constants:

$\ln 2 + 1 - \ln x = 1 + \ln 2 - \ln x$

Conclusion

After applying the Quotient Rule, Product Rule, and simplifying using the properties of logarithms, we have simplified the given expression to:

$1 + \ln 2 - \ln x$

This expression is equivalent to option A.

The Correct Answer

The correct answer is:

A. $1 + \ln 2 - \ln x$

Discussion

This problem requires a clear understanding of the properties of logarithms and the ability to apply them to simplify complex expressions. By breaking down the expression into manageable parts and using the Quotient Rule, Product Rule, and simplifying using the properties of logarithms, we can arrive at the correct equivalent expression.

Additional Tips and Tricks

• When simplifying logarithmic expressions, always start by applying the Quotient Rule and then the Product Rule.
• Use the properties of logarithms to simplify the expression further.
• Be careful when simplifying expressions involving constants, as they may affect the final result.

Introduction

Logarithmic expressions can be complex and challenging to simplify. However, with a clear understanding of the properties of logarithms, we can break down these expressions into more manageable parts. In this article, we will explore some common questions and answers related to logarithmic expressions.

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

A: A logarithmic expression is the inverse of an exponential expression. While an exponential expression represents a power or an exponent, a logarithmic expression represents the power or exponent that a base must be raised to in order to produce a given value.

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, such as the Product Rule and the Quotient Rule. For example, if you have the expression $\log_b (xy)$, you can simplify it using the Product Rule as $\log_b x + \log_b y$.

Q: What is the logarithmic property that states $\log_b (x^y) = y \log_b x$?

A: This property is known as the Power Rule. It states that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base.

Q: How do I evaluate a logarithmic expression with a negative exponent?

A: To evaluate a logarithmic expression with a negative exponent, you can use the property $\log_b (x^{-y}) = -y \log_b x$. For example, if you have the expression $\log_2 (x^{-3})$, you can simplify it as $-3 \log_2 x$.

Q: What is the difference between a common logarithm and a natural logarithm?

A: A common logarithm is a logarithm with a base of 10, while a natural logarithm is a logarithm with a base of e. The natural logarithm is often denoted as $\ln x$.

Q: How do I convert a common logarithm to a natural logarithm?

A: To convert a common logarithm to a natural logarithm, you can use the property $\log_{10} x = \frac{\ln x}{\ln 10}$. For example, if you have the expression $\log_{10} x$, you can convert it to a natural logarithm as $\frac{\ln x}{\ln 10}$.

Q: What is the logarithmic property that states $\log_b (x) + \log_b (y) = \log_b (xy)$?

A: This property is known as the Product Rule. It states that the logarithm of a product is equal to the sum of the logarithms of the individual terms.

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 $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$. For example, if you have the expression $\log_2 \left(\frac{x}{y}\right)$, you can simplify it as $\log_2 x - \log_2 y$.

Conclusion

Logarithmic expressions can be complex and challenging to simplify. However, with a clear understanding of the properties of logarithms, we can break down these expressions into more manageable parts. By following the tips and tricks outlined in this article, you can become more confident in simplifying logarithmic expressions and identifying the correct equivalent expression from a list of options.

Additional Tips and Tricks

• When simplifying logarithmic expressions, always start by applying the Quotient Rule and then the Product Rule.
• Use the properties of logarithms to simplify the expression further.
• Be careful when simplifying expressions involving constants, as they may affect the final result.
• Practice, practice, practice! The more you practice simplifying logarithmic expressions, the more confident you will become.