https://file.aiquickdraw.com/mathbot/file/1725737858191p5rnkayv.jpg

It appears that I couldn't extract the text successfully from the image due to its quality or format. Let me manually interpret the mathematical expression based on the image.

From what I can see, the equation is:

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

The goal is to simplify this equation into a single logarithmic expression.

### Step-by-Step Solution:

1. **Simplify both sides**: First, we notice that both sides of the equation have \(\frac{1}{2}\). So, we can cancel \(\frac{1}{2}\) from both sides.

   \[
   \ln(x + 2)^{2} = \ln|x| - \ln(x^{2} + 3x - 2)
   \]

2. **Use logarithmic properties**: Apply the logarithmic rule \(\ln(a^{b}) = b \ln(a)\) on the left-hand side.

   \[
   2 \ln(x + 2) = \ln|x| - \ln(x^{2} + 3x - 2)
   \]

3. **Combine logarithms**: Use the property of logarithms \(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\) on the right-hand side.

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

4. **Exponentiate both sides**: Now, exponentiate both sides to get rid of the natural logarithm.

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

   Simplify:

   \[
   (x + 2)^{2} = \frac{|x|}{x^{2} + 3x - 2}
   \]

Thus, the expression is simplified to:

\[
(x + 2)^{2} = \frac{|x|}{x^{2} + 3x - 2}
\]

Let me know if you would like more details on any part or have questions!

### Related Questions:
1. What are the basic logarithmic properties?
2. How do you solve logarithmic equations with different bases?
3. What are the common logarithmic simplifications?
4. What happens if \(x\) is negative in a logarithmic equation?
5. How does exponentiation work to remove logarithms?

### Tip:
Always remember to check the domain of logarithmic expressions—especially when dealing with \(\ln(x)\)—as logarithms are only defined for positive values of \(x\).

Simplify the expression ½ln(x+2)² = ½[ln|x| - ln(x² + 3x - 2)] into a single logarithm.

Learn how to simplify the logarithmic expression ½ln(x+2)² = ½[ln|x| - ln(x² + 3x - 2)] into a single logarithmic expression. This problem covers key algebraic and logarithmic concepts suitable for high school students in Grades 11-12.

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