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 $x$.