The expression you have is:

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

Let's break it down:

Step 1: Simplify the logarithmic expression

Using logarithmic rules, we can simplify the expression:

1. Logarithm of a quotient: $\log\left(\frac{a}{b}\right) = \log(a) - \log(b)$

   Applying this rule to the given expression: [ \log\left(\frac{x^2}{\sqrt[3]{(x+8)^5}}\right) = \log(x^2) - \log\left(\sqrt[3]{(x+8)^5}\right) ]

Step 2: Simplify each term

2. Logarithm of a power: $\log(a^b) = b \log(a)$

   So for the first term: $\log(x^2) = 2 \log(x)$

   Now, simplifying the second term: $\log\left(\sqrt[3]{(x+8)^5}\right) = \log\left((x+8)^{5/3}\right) = \frac{5}{3} \log(x+8)$

Step 3: Combine the results

The expression now becomes: $2 \log(x) - \frac{5}{3} \log(x+8)$

Thus, the simplified form of the original expression is: $2 \log(x) - \frac{5}{3} \log(x+8)$

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Would you like further explanation or help with similar problems? Here are some related questions:

1. How do logarithmic rules apply when simplifying logarithms with different bases?
2. What happens if the logarithm is of a product instead of a quotient?
3. Can you simplify expressions with logarithms involving exponents greater than 2 or fractions?
4. How do you handle logarithmic expressions with radicals in the denominator?
5. How can this type of simplification be applied to more complex logarithmic equations?

Tip: Always remember that when simplifying logarithmic expressions, you can use the logarithmic properties such as the product rule, quotient rule, and power rule to break down complex expressions.