Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.
\[
\log \left[\frac{10 x^{4} \sqrt[3]{6-x}}{7(x+6)^{2}}\right]
\]
\[
\log \left[\frac{10 x^{4} \sqrt[3]{6-x}}{7(x+6)^{2}}\right]=\square
\]

Step 1 ：The given expression is a logarithm of a fraction. We can use the properties of logarithms to simplify this expression. The properties we will use are: \(\log(a/b) = \log(a) - \log(b)\), \(\log(a^n) = n \cdot \log(a)\), and \(\log(\sqrt[n]{a}) = \frac{1}{n} \cdot \log(a)\).

Step 2 ：We can apply these properties step by step to simplify the given expression.

Step 3 ：Using the properties of logarithms, the given expression \(\log \left[\frac{10 x^{4} \sqrt[3]{6-x}}{7(x+6)^{2}}\right]\) can be expanded as \(4\log(x) + \frac{1}{3}\log(6 - x) - 2\log(x + 6) - \log(7) + \log(10)\).

Step 4 ：\(\boxed{\log \left[\frac{10 x^{4} \sqrt[3]{6-x}}{7(x+6)^{2}}\right]=4\log(x) + \frac{1}{3}\log(6 - x) - 2\log(x + 6) - \log(7) + \log(10)}\)