Answer
The equivalent expression for $\frac{1}{3}\left[ {{\log }_{a}}x-2{{\log }_{a}}y \right]$ is ${{\log }_{a}}\sqrt[3]{\left( \frac{x}{{{y}^{2}}} \right)}$.
Work Step by Step
$\frac{1}{3}\left[ {{\log }_{a}}x-2{{\log }_{a}}y \right]$
Apply the power rule for logarithms as follows.
$\frac{1}{3}\left[ {{\log }_{a}}x-2{{\log }_{a}}y \right]=\frac{1}{3}\left[ {{\log }_{a}}x-{{\log }_{a}}{{y}^{2}} \right]$
Apply the quotient rule for logarithms as follows.
$\frac{1}{3}\left[ {{\log }_{a}}x-2{{\log }_{a}}y \right]=\frac{1}{3}\left( {{\log }_{a}}\frac{x}{{{y}^{2}}} \right)$
Apply the power rule for logarithms as follows.
$\begin{align}
& \frac{1}{3}\left[ {{\log }_{a}}x-2{{\log }_{a}}y \right]={{\log }_{a}}{{\left( \frac{x}{{{y}^{2}}} \right)}^{\frac{1}{3}}} \\
& ={{\log }_{a}}\sqrt[3]{\left( \frac{x}{{{y}^{2}}} \right)}
\end{align}$
