The problem asks to write the expression $\frac{1}{2} [4 \log x - \log \sqrt{x^5}] + \log \sqrt[3]{x^2}$ as a single logarithm.

1. Problem Description

The problem asks to write the expression $\frac{1}{2} [4 \log x - \log \sqrt{x^5}] + \log \sqrt[3]{x^2}$ as a single logarithm.

2. Solution Steps

We are given the expression $\frac{1}{2} [4 \log x - \log \sqrt{x^5}] + \log \sqrt[3]{x^2}$. First, we simplify the terms inside the brackets:

$\sqrt{x^5} = x^{\frac{5}{2}}$ and $\sqrt[3]{x^2} = x^{\frac{2}{3}}$

So, $\log \sqrt{x^5} = \log x^{\frac{5}{2}} = \frac{5}{2} \log x$ and $\log \sqrt[3]{x^2} = \log x^{\frac{2}{3}} = \frac{2}{3} \log x$.

Now substitute these values back into the expression:

$\frac{1}{2} [4 \log x - \frac{5}{2} \log x] + \frac{2}{3} \log x$

$\frac{1}{2} [\frac{8}{2} \log x - \frac{5}{2} \log x] + \frac{2}{3} \log x$

$\frac{1}{2} [\frac{3}{2} \log x] + \frac{2}{3} \log x$

$\frac{3}{4} \log x + \frac{2}{3} \log x$

Now, find a common denominator for the fractions:

$(\frac{3}{4} + \frac{2}{3}) \log x$

$(\frac{9}{12} + \frac{8}{12}) \log x$

$\frac{17}{12} \log x$

3. Final Answer

The final answer is $\frac{17}{12} \log x$.