The expression you provided is:

$\log \left( \frac{6(x+5)^2}{\sqrt[3]{x^4}} \right)$

Let's simplify this step by step.

Step 1: Apply logarithmic properties

We can break this down using the property of logarithms that states:

$\log \left( \frac{A}{B} \right) = \log(A) - \log(B)$

Thus,

[ \log \left( \frac{6(x+5)^2}{\sqrt[3]{x^4}} \right) = \log(6(x+5)^2) - \log\left(\sqrt[3]{x^4}\right) ]

Step 2: Simplify the logarithms further

For the first term $\log(6(x+5)^2)$:

Using the product rule for logarithms, $\log(AB) = \log(A) + \log(B)$:

$\log(6(x+5)^2) = \log(6) + \log((x+5)^2)$

Now, apply the power rule, $\log(A^n) = n\log(A)$:

$\log((x+5)^2) = 2\log(x+5)$

Thus, the first part becomes:

$\log(6(x+5)^2) = \log(6) + 2\log(x+5)$

For the second term $\log\left(\sqrt[3]{x^4}\right)$:

Recall that $\sqrt[3]{x^4} = x^{4/3}$, and apply the power rule:

$\log\left(\sqrt[3]{x^4}\right) = \log(x^{4/3}) = \frac{4}{3}\log(x)$

Step 3: Combine the results

Now combine everything:

$\log \left( \frac{6(x+5)^2}{\sqrt[3]{x^4}} \right) = \log(6) + 2\log(x+5) - \frac{4}{3}\log(x)$

This is the simplified form of the expression.

Would you like any more details on this, or do you have any questions?

Here are some related questions you might find helpful:

1. How can logarithmic rules help simplify more complex expressions?
2. How does the power rule apply to logarithms in general?
3. What are some common mistakes when working with logarithmic expressions?
4. How do fractional powers, like $x^{4/3}$, affect logarithmic simplifications?
5. How can you expand logarithmic expressions with multiple variables?

Tip: Always remember to apply the logarithmic properties in a step-by-step manner for clear simplification!