The given expression is:

$\log \left( \sqrt[3]{\frac{(x+8)^2}{x^5}} \right)$

Simplification Steps:

1. Apply the cube root:

   • The cube root of a fraction is the cube root of the numerator and denominator separately: [ \sqrt[3]{\frac{(x+8)^2}{x^5}} = \frac{\sqrt[3]{(x+8)^2}}{\sqrt[3]{x^5}} ]

2. Simplify the powers under the cube root:

   • The cube root of $(x+8)^2$ is $(x+8)^{\frac{2}{3}}$, and the cube root of $x^5$ is $x^{\frac{5}{3}}$: $\frac{(x+8)^{\frac{2}{3}}}{x^{\frac{5}{3}}}$

3. Use logarithm properties:

   • Using the logarithmic property $\log \left( \frac{a}{b} \right) = \log a - \log b$: $\log \left( \frac{(x+8)^{\frac{2}{3}}}{x^{\frac{5}{3}}} \right) = \log \left( (x+8)^{\frac{2}{3}} \right) - \log \left( x^{\frac{5}{3}} \right)$

4. Simplify the logs using $\log(a^b) = b \log a$: $\frac{2}{3} \log (x+8) - \frac{5}{3} \log x$

Thus, the simplified form of the expression is:

$\frac{2}{3} \log (x+8) - \frac{5}{3} \log x$

Would you like further details or have any questions about these steps?

Here are some related questions:

1. What are the other logarithmic properties and how are they used in solving equations?
2. How do you handle logarithms of different bases in algebraic expressions?
3. Can you explain the importance of simplifying logarithmic expressions in real-life problems?
4. How does the power rule of logarithms affect solving exponential equations?
5. What is the significance of the cube root in logarithmic functions?

Tip: Logarithmic properties are essential for simplifying complex expressions and solving equations efficiently.