1. **State the problem:** Simplify the expression $\log \frac{x^4}{\sqrt[3]{y^5 z^5}}$. 2. **Recall the logarithm rule:** $\log \frac{a}{b} = \log a - \log b$. 3. **Apply the rule:** $$\log \frac{x^4}{\sqrt[3]{y^5 z^5}} = \log x^4 - \log \sqrt[3]{y^5 z^5}$$ 4. **Use the power rule for logarithms:** $\log a^m = m \log a$. 5. **Simplify each term:** $$\log x^4 = 4 \log x$$ $$\log \sqrt[3]{y^5 z^5} = \log (y^5 z^5)^{\frac{1}{3}} = \frac{1}{3} \log (y^5 z^5)$$ 6. **Apply the product rule for logarithms:** $\log (ab) = \log a + \log b$. 7. **Simplify further:** $$\frac{1}{3} \log (y^5 z^5) = \frac{1}{3} (\log y^5 + \log z^5) = \frac{1}{3} (5 \log y + 5 \log z) = \frac{5}{3} \log y + \frac{5}{3} \log z$$ 8. **Combine all parts:** $$4 \log x - \left( \frac{5}{3} \log y + \frac{5}{3} \log z \right) = 4 \log x - \frac{5}{3} \log y - \frac{5}{3} \log z$$ **Final answer:** $$\boxed{4 \log x - \frac{5}{3} \log y - \frac{5}{3} \log z}$$