1. **State the problem:** Expand the logarithm $$\log_3 \left( \frac{x^4}{\sqrt[3]{y^5 z^5}} \right)$$ fully using properties of logarithms, expressing the answer in terms of $$\log x$$, $$\log y$$, and $$\log z$$. 2. **Recall the logarithm properties:** - $$\log_b \left( \frac{A}{B} \right) = \log_b A - \log_b B$$ - $$\log_b (A^r) = r \log_b A$$ - $$\log_b (AB) = \log_b A + \log_b B$$ 3. **Apply the quotient rule:** $$\log_3 \left( \frac{x^4}{\sqrt[3]{y^5 z^5}} \right) = \log_3 (x^4) - \log_3 \left( (y^5 z^5)^{\frac{1}{3}} \right)$$ 4. **Apply the power rule to numerator:** $$\log_3 (x^4) = 4 \log_3 x$$ 5. **Simplify the denominator inside the logarithm:** $$(y^5 z^5)^{\frac{1}{3}} = y^{\frac{5}{3}} z^{\frac{5}{3}}$$ 6. **Apply the product rule to denominator:** $$\log_3 \left( y^{\frac{5}{3}} z^{\frac{5}{3}} \right) = \log_3 \left( y^{\frac{5}{3}} \right) + \log_3 \left( z^{\frac{5}{3}} \right)$$ 7. **Apply the power rule to each term:** $$\log_3 \left( y^{\frac{5}{3}} \right) = \frac{5}{3} \log_3 y$$ $$\log_3 \left( z^{\frac{5}{3}} \right) = \frac{5}{3} \log_3 z$$ 8. **Combine all parts:** $$4 \log_3 x - \left( \frac{5}{3} \log_3 y + \frac{5}{3} \log_3 z \right) = 4 \log_3 x - \frac{5}{3} \log_3 y - \frac{5}{3} \log_3 z$$ **Final answer:** $$\boxed{4 \log_3 x - \frac{5}{3} \log_3 y - \frac{5}{3} \log_3 z}$$