1. **State the problem:** Expand the logarithmic expression $$\log \left( \sqrt[3]{\frac{(x + 8)^2}{x^5}} \right)$$ using properties of logarithms. 2. **Recall the properties of logarithms:** - $$\log(a^b) = b \log(a)$$ - $$\log\left(\frac{a}{b}\right) = \log(a) - \log(b)$$ - The cube root can be written as a fractional exponent: $$\sqrt[3]{A} = A^{\frac{1}{3}}$$ 3. **Rewrite the expression using fractional exponents:** $$\log \left( \left( \frac{(x + 8)^2}{x^5} \right)^{\frac{1}{3}} \right)$$ 4. **Apply the power rule:** $$= \frac{1}{3} \log \left( \frac{(x + 8)^2}{x^5} \right)$$ 5. **Apply the quotient rule:** $$= \frac{1}{3} \left( \log \left( (x + 8)^2 \right) - \log \left( x^5 \right) \right)$$ 6. **Apply the power rule inside the logarithms:** $$= \frac{1}{3} \left( 2 \log (x + 8) - 5 \log x \right)$$ 7. **Distribute the $$\frac{1}{3}$$:** $$= \frac{2}{3} \log (x + 8) - \frac{5}{3} \log x$$ **Final expanded form:** $$\log \left( \sqrt[3]{\frac{(x + 8)^2}{x^5}} \right) = \frac{2}{3} \log (x + 8) - \frac{5}{3} \log x$$ This expression has no radicals or exponents inside the logarithms, as requested.