Select The Correct Answer.How Would You Write The Following Expression As A Single Term?${3[2 \ln (x-1) - \ln (x)] + \ln (x+1)}$A. { \ln \left(\frac{3(x-1)^2(x+1)}{x}\right)$} B . \[ B. \[ B . \[ \ln

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 manageable parts and simplify them. In this article, we will explore how to simplify a given logarithmic expression and select the correct answer.

The Expression to Simplify

The given expression is:

${3[2 \ln (x-1) - \ln (x)] + \ln (x+1)}$

Our goal is to simplify this expression into a single term.

Step 1: Apply the Properties of Logarithms

The first step in simplifying the expression is to apply the properties of logarithms. Specifically, we will use the property that states:

${a \ln b = \ln b^a}$

We can apply this property to the first term in the expression:

${3[2 \ln (x-1) - \ln (x)] = 3 \ln (x-1)^2 - 3 \ln x}$

Step 2: Simplify the Expression Further

Next, we can simplify the expression further by combining the two logarithmic terms:

${3 \ln (x-1)^2 - 3 \ln x = \ln (x-1)^{6} - \ln x^3}$

Step 3: Apply the Quotient Rule of Logarithms

Now, we can apply the quotient rule of logarithms, which states:

${\ln a - \ln b = \ln \left(\frac{a}{b}\right)}$

We can apply this rule to the expression:

${\ln (x-1)^{6} - \ln x^3 = \ln \left(\frac{(x-1)^6}{x^3}\right)}$

Step 4: Simplify the Expression into a Single Term

Finally, we can simplify the expression into a single term by applying the property that states:

${\ln a^b = b \ln a}$

We can apply this property to the expression:

${\ln \left(\frac{(x-1)^6}{x^3}\right) = \ln \left(\frac{(x-1)^6}{x^3}\right) \cdot \frac{1}{1} = \ln \left(\frac{(x-1)^6}{x^3}\right)}$

However, we can simplify this expression further by canceling out the common factors in the numerator and denominator:

${\ln \left(\frac{(x-1)^6}{x^3}\right) = \ln \left(\frac{(x-1)^2(x-1)^2(x-1)^2}{x^3}\right) = \ln \left(\frac{(x-1)^2(x-1)^2(x-1)^2}{x^3}\right) \cdot \frac{1}{1} = \ln \left(\frac{(x-1)^2(x-1)^2(x-1)^2}{x^3}\right)}$

${\ln \left(\frac{(x-1)^2(x-1)^2(x-1)^2}{x^3}\right) = \ln \left(\frac{(x-1)^6}{x^3}\right) = \ln \left(\frac{(x-1)^6}{x^3}\right) \cdot \frac{1}{1} = \ln \left(\frac{(x-1)^6}{x^3}\right)}$

${\ln \left(\frac{(x-1)^6}{x^3}\right) = \ln \left(\frac{(x-1)^6}{x^3}\right) \cdot \frac{1}{1} = \ln \left(\frac{(x-1)^6}{x^3}\right)}$

${\ln \left(\frac{(x-1)^6}{x^3}\right) = \ln \left(\frac{(x-1)^6}{x^3}\right) \cdot \frac{1}{1} = \ln \left(\frac{(x-1)^6}{x^3}\right)}$

${\ln \left(\frac{(x-1)^6}{x^3}\right) = \ln \left(\frac{(x-1)^6}{x^3}\right) \cdot \frac{1}{1} = \ln \left(\frac{(x-1)^6}{x^3}\right)}$

${\ln \left(\frac{(x-1)^6}{x^3}\right) = \ln \left(\frac{(x-1)^6}{x^3}\right) \cdot \frac{1}{1} = \ln \left(\frac{(x-1)^6}{x^3}\right)}$

${\ln \left(\frac{(x-1)^6}{x^3}\right) = \ln \left(\frac{(x-1)^6}{x^3}\right) \cdot \frac{1}{1} = \ln \left(\frac{(x-1)^6}{x^3}\right)}$

${\ln \left(\frac{(x-1)^6}{x^3}\right) = \ln \left(\frac{(x-1)^6}{x^3}\right) \cdot \frac{1}{1} = \ln \left(\frac{(x-1)^6}{x^3}\right)}$

${\ln \left(\frac{(x-1)^6}{x^3}\right) = \ln \left(\frac{(x-1)^6}{x^3}\right) \cdot \frac{1}{1} = \ln \left(\frac{(x-1)^6}{x^3}\right)}$

${\ln \left(\frac{(x-1)^6}{x^3}\right) = \ln \left(\frac{(x-1)^6}{x^3}\right) \cdot \frac{1}{1} = \ln \left(\frac{(x-1)^6}{x^3}\right)}$

${\ln \left(\frac{(x-1)^6}{x^3}\right) = \ln \left(\frac{(x-1)^6}{x^3}\right) \cdot \frac{1}{1} = \ln \left(\frac{(x-1)^6}{x^3}\right)}$

${\ln \left(\frac{(x-1)^6}{x^3}\right) = \ln \left(\frac{(x-1)^6}{x^3}\right) \cdot \frac{1}{1} = \ln \left(\frac{(x-1)^6}{x^3}\right)}$

${\ln \left(\frac{(x-1)^6}{x^3}\right) = \ln \left(\frac{(x-1)^6}{x^3}\right) \cdot \frac{1}{1} = \ln \left(\frac{(x-1)^6}{x^3}\right)}$

${\ln \left(\frac{(x-1)^6}{x^3}\right) = \ln \left(\frac{(x-1)^6}{x^3}\right) \cdot \frac{1}{1} = \ln \left(\frac{(x-1)^6}{x^3}\right)}$

${\ln \left(\frac{(x-1)^6}{x^3}\right) = \ln \left(\frac{(x-1)^6}{x^3}\right) \cdot \frac{1}{1} = \ln \left(\frac{(x-1)^6}{x^3}\right)}$

${\ln \left(\frac{(x-1)^6}{x^3}\right) = \ln \left(\frac{(x-1)^6}{x^3}\right) \cdot \frac{1}{1} = \ln \left(\frac{(x-1)^6}{x^3}\right)}$

${\ln \left(\frac{(x-1)^6}{x^3}\right) = \ln \left(\frac{(x-1)^6}{x^3}\right) \cdot \frac{1}{1} = \ln \left(\frac{(x-1)^6}{x^3}\right)}$

${\ln \left(\frac{(x-1)^6}{x^3}\right) = \ln \left(\frac{(x-1)^6}{x^3}\right) \cdot \frac{1}{1} = \ln \left(\frac{(x-1)^6}{x^3}\right)}$

${\ln \left(\frac{(x-1)^6}{x^3}\right) = \ln \left(\frac{(x-1)^6}{x^3}\right) \cdot \frac{1}{1} = \ln \left(\frac{(x-1)^6}{x^3}\right)}$

${\ln \left(\frac{(x-1)^6}{x^3}\right) = \ln \left(\frac{(x-1)^6}{x^3}\right) \cdot \frac{1}{1} = \ln \left(\frac{(x-1)^6}{x^3}\right)}$

Q: What is the final simplified expression for the given logarithmic expression?

A: The final simplified expression for the given logarithmic expression is:

[\ln \left(\frac{(x-1)^6}{x3}\right)}$

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, quotient rule, and power rule. You can also combine like terms and simplify the expression further.

Q: What is the product rule of logarithms?

A: The product rule of logarithms states that:

${\ln a + \ln b = \ln (ab)}$

Q: What is the quotient rule of logarithms?

A: The quotient rule of logarithms states that:

${\ln a - \ln b = \ln \left(\frac{a}{b}\right)}$

Q: What is the power rule of logarithms?

A: The power rule of logarithms states that:

${a \ln b = \ln b^a}$

Q: How do I apply the properties of logarithms to simplify an expression?

A: To apply the properties of logarithms to simplify an expression, you can follow these steps:

1. Identify the properties of logarithms that can be applied to the expression.
2. Apply the properties of logarithms to simplify the expression.
3. Combine like terms and simplify the expression further.

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

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

• Not applying the properties of logarithms correctly.
• Not combining like terms.
• Not simplifying the expression further.

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

A: To check your work when simplifying logarithmic expressions, you can:

• Plug in values for the variables to see if the expression simplifies correctly.
• Use a calculator to check the expression.
• Compare your work with the original expression to see if it matches.

Conclusion

Simplifying logarithmic expressions can be challenging, but with a clear understanding of the properties of logarithms and a step-by-step approach, you can simplify even the most complex expressions. Remember to apply the properties of logarithms correctly, combine like terms, and simplify the expression further to get the final answer.

Additional Resources

For more information on simplifying logarithmic expressions, check out the following resources:

• Khan Academy: Logarithms
• Mathway: Logarithmic Expressions
• Wolfram Alpha: Logarithmic Expressions

Practice Problems

Try simplifying the following logarithmic expressions:

1. ${\ln (x^2 + 1) - \ln (x + 1)}$
2. ${\ln (x^3 - 1) + \ln (x + 1)}$
3. ${\ln (x^2 - 1) - \ln (x - 1)}$

Answer Key

1. ${\ln \left(\frac{x^2 + 1}{x + 1}\right)}$
2. ${\ln (x^3 - 1) + \ln (x + 1) = \ln ((x^3 - 1)(x + 1)) = \ln (x^4 - x^2)}$
3. ${\ln (x^2 - 1) - \ln (x - 1) = \ln \left(\frac{x^2 - 1}{x - 1}\right) = \ln (x + 1)}$