https://file.aiquickdraw.com/mathbot/file/1725737776809o6kwllg8.jpg

To rewrite the given expression as a single logarithm, let's first clarify the expression that is shown in the image:

\[
\frac{2}{5} \ln(x + 2)^5 + \frac{1}{2} \left[ \ln(x^4) - \ln(x^2 - 3x + 2)^2 \right]
\]

### Step-by-Step Solution

1. **Simplify the expression inside the brackets:**

   Using the logarithm rule \(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\), we have:

   \[
   \ln(x^4) - \ln(x^2 - 3x + 2)^2 = \ln\left(\frac{x^4}{(x^2 - 3x + 2)^2}\right)
   \]

2. **Multiply by the coefficient \(\frac{1}{2}\):**

   Apply the coefficient \(\frac{1}{2}\) to the logarithm:

   \[
   \frac{1}{2} \ln\left(\frac{x^4}{(x^2 - 3x + 2)^2}\right) = \ln\left(\left(\frac{x^4}{(x^2 - 3x + 2)^2}\right)^{1/2}\right)
   \]

   This simplifies to:

   \[
   \ln\left(\frac{x^2}{x^2 - 3x + 2}\right)
   \]

3. **Simplify the first term:**

   Using the logarithm power rule \(a \ln(b) = \ln(b^a)\), we can rewrite the first term:

   \[
   \frac{2}{5} \ln(x + 2)^5 = \ln\left((x + 2)^2\right)
   \]

4. **Combine the logarithms:**

   Now, combine the two logarithms into a single logarithm using the addition rule \(\ln(a) + \ln(b) = \ln(ab)\):

   \[
   \ln\left((x + 2)^2 \cdot \frac{x^2}{x^2 - 3x + 2}\right)
   \]

   This simplifies to:

   \[
   \ln\left(\frac{(x + 2)^2 \cdot x^2}{x^2 - 3x + 2}\right)
   \]

### Final Answer

The given expression rewritten as a single logarithm is:

\[
\ln\left(\frac{(x + 2)^2 \cdot x^2}{x^2 - 3x + 2}\right)
\]

Would you like more details on this solution or

Rewrite the expression as a single logarithm: \frac{2}{5} \ln(x + 2)^5 + \frac{1}{2} \left[ \ln(x^4) - \ln(x^2 - 3x + 2)^2 \right]

This detailed solution demonstrates how to simplify a logarithmic expression involving multiple terms using logarithmic properties. It covers key algebraic concepts, including the subtraction, addition, and power rules for logarithms. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation