1. The problem is to simplify the expression $\frac{1}{4} \log \frac{1}{4}$. 2. Recall the logarithm power rule: $a \log b = \log b^a$. We can rewrite the expression as $\log \left(\frac{1}{4}\right)^{\frac{1}{4}}$. 3. Calculate the exponentiation: $\left(\frac{1}{4}\right)^{\frac{1}{4}} = \frac{1^{\frac{1}{4}}}{4^{\frac{1}{4}}} = \frac{1}{4^{\frac{1}{4}}}$. 4. Since $4 = 2^2$, then $4^{\frac{1}{4}} = (2^2)^{\frac{1}{4}} = 2^{\frac{2}{4}} = 2^{\frac{1}{2}} = \sqrt{2}$. 5. Therefore, $\log \left(\frac{1}{4}\right)^{\frac{1}{4}} = \log \frac{1}{\sqrt{2}}$. 6. This is the simplified form of the original expression. Final answer: $\frac{1}{4} \log \frac{1}{4} = \log \frac{1}{\sqrt{2}}$