Question 6 Write the expression as a sum and/or difference of logarithms. Express powers as factors. \[ \ln \left[\frac{x^{2}-x-20}{(x+7)^{4}}\right]^{1 / 5}, x>5 \] $\ln \left[\frac{x^{2}-x-20}{(x+7)^{4}}\right]^{1 / 5}=\square$ (Simplify your answer. Type an exact answer. Use integers or fractions for any numbers in the expression.)

Step 1 ：Given the expression \(\ln \left[\frac{x^{2}-x-20}{(x+7)^{4}}\right]^{1 / 5}\), we can apply the properties of logarithms to simplify it.

Step 2 ：First, we can use the power rule of logarithms, which states that \(\ln a^{n} = n \ln a\), to rewrite the expression as \(\frac{1}{5} \ln (x^{2}-x-20) - \frac{4}{5} \ln (x+7)\).

Step 3 ：Next, we can use the quotient rule of logarithms, which states that \(\ln \frac{a}{b} = \ln a - \ln b\), to rewrite the expression as \(\frac{1}{5} \ln x^{2} - \frac{1}{5} \ln x - \frac{1}{5} \ln 20 - \frac{4}{5} \ln (x+7)\).

Step 4 ：So, the final answer is \(\boxed{\frac{1}{5} \ln x^{2} - \frac{1}{5} \ln x - \frac{1}{5} \ln 20 - \frac{4}{5} \ln (x+7)}\).