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