Answer
The equivalent expression for ${{\log }_{a}}\frac{{{x}^{5}}}{y{{z}^{2}}}$ is $5{{\log }_{a}}x-{{\log }_{a}}y-2{{\log }_{a}}z$.
Work Step by Step
${{\log }_{a}}\frac{{{x}^{5}}}{y{{z}^{2}}}$
Apply the quotient rule for logarithms as follows.
$\begin{align}
& {{\log }_{a}}\frac{{{x}^{5}}}{y{{z}^{2}}}={{\log }_{a}}{{x}^{5}}-{{\log }_{a}}y{{z}^{2}} \\
& ={{\log }_{a}}{{x}^{5}}-{{\log }_{a}}y-{{\log }_{a}}{{z}^{2}} \\
& ={{\log }_{a}}{{x}^{5}}-\left( {{\log }_{a}}y+{{\log }_{a}}{{z}^{2}} \right)
\end{align}$
Apply the power rule for logarithm as follows.
${{\log }_{a}}\frac{{{x}^{5}}}{y{{z}^{2}}}=5{{\log }_{a}}x-{{\log }_{a}}y-2{{\log }_{a}}z$
