Answer
The required solution is ${{\log }_{7}}\left( \frac{\sqrt[5]{x}}{49{{y}^{10}}} \right)=\frac{1}{5}{{\log }_{7}}x-2-10{{\log }_{7}}y $.
Work Step by Step
$\begin{align}
& {{\log }_{7}}\left( \frac{{{x}^{\frac{1}{5}}}}{49{{y}^{10}}} \right)={{\log }_{7}}{{x}^{\frac{1}{5}}}-{{\log }_{7}}\left( 49{{y}^{10}} \right) \\
& ={{\log }_{7}}{{x}^{\frac{1}{5}}}-\left( {{\log }_{7}}49+10{{\log }_{7}}y \right) \\
& =\frac{1}{5}{{\log }_{7}}x-{{\log }_{7}}49-10{{\log }_{7}}y
\end{align}$
${{\log }_{7}}49=2$ because 2 is the power to which 7 must be raised to get 49.
Thus,
${{\log }_{7}}\left( \frac{\sqrt[5]{x}}{49{{y}^{10}}} \right)=\frac{1}{5}{{\log }_{7}}x-2-10{{\log }_{7}}y $.
