Use the properties of logarithms to expand the following expression.
\[
\log \left(\sqrt[3]{\frac{(x+5)^{4}}{x^{2}}}\right)
\]

Your answer should not have radicals or exponents.
You may assume that all variables are positive.
\[
\log \left(\sqrt[3]{\frac{(x+5)^{4}}{x^{2}}}\right)=\square
\]
$\square \log$

Step 1 ：Expand the expression using the property of logarithms: \(\log \left(\sqrt[3]{\frac{(x+5)^{4}}{x^{2}}}\right) = \frac{1}{3} \log \left(\frac{(x+5)^{4}}{x^{2}}\right)\)

Step 2 ：Use the property of logarithms that allows us to write the logarithm of a quotient as the difference of the logarithms: \(\frac{1}{3} \log \left(\frac{(x+5)^{4}}{x^{2}}\right) = \frac{1}{3} \left( \log (x+5)^{4} - \log x^{2} \right)\)

Step 3 ：Use the property of logarithms that allows us to bring the exponent in front of the logarithm: \(\frac{1}{3} \left( \log (x+5)^{4} - \log x^{2} \right) = \frac{4}{3} \log (x+5) - \frac{2}{3} \log x\)

Step 4 ：Final Answer: \(\boxed{\log \left(\sqrt[3]{\frac{(x+5)^{4}}{x^{2}}}\right) = \frac{4}{3} \log (x+5) - \frac{2}{3} \log x}\)