The expression you provided is:

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

To simplify this logarithmic expression, we can use the **properties of logarithms**. Specifically, the properties we'll use are:

1. \( a \ln(b) = \ln(b^a) \) — logarithm power rule
2. \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \) — logarithm subtraction rule

### Step-by-step solution:

1. Apply the power rule to the first term:
   \[
   3 \ln(x + 2) = \ln((x + 2)^3)
   \]

   So the expression becomes:
   \[
   \ln((x + 2)^3) - \ln(x - 1)
   \]

2. Apply the logarithmic subtraction rule:
   \[
   \ln\left(\frac{(x + 2)^3}{x - 1}\right)
   \]

Thus, the simplified form of the expression is:
\[
\ln\left(\frac{(x + 2)^3}{x - 1}\right)
\]

This is the final simplified expression.

Would you like more details on any step or have any questions?

Here are some related questions to explore:
1. What is the derivative of this logarithmic expression?
2. How do you simplify logarithmic expressions with more complex coefficients?
3. What happens if \( x = 1 \) in this expression?
4. How does the logarithmic function behave when applied to negative inputs?
5. Can this expression be expanded back to its original form?

**Tip:** Always remember to check the domain of logarithmic functions. Here, \( x > 1 \), since both \( x + 2 \) and \( x - 1 \) must be positive.

Discover how to simplify the logarithmic expression 3ln(x + 2) - ln(x - 1) using the properties of logarithms. This step-by-step guide explains key concepts like the power rule and subtraction rule, perfect for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation