Answer
$\frac{1}{3}log_{10}x-\frac{1}{3}log_{10}y$
Work Step by Step
We know that $log(x)$ is a common logarithm with an understand base of 10. Therefore, $log\sqrt[3] \frac{x}{y}=log_{10}\sqrt[3] \frac{x}{y}=log_{10}\frac{\sqrt[3] x}{\sqrt[3] y}$.
Based on the quotient rule of logarithms, we know that $log_{b}(\frac{M}{N})=log_{b}M-log_{b}N$ (where $b$, $M$, and $N$ are positive real numbers and $b\ne1$).
Therefore, $log_{10}\frac{\sqrt[3] x}{\sqrt[3] y}=log_{10}\sqrt[3] x-log_{10}\sqrt[3] y=log_{10}x^{\frac{1}{3}}-log_{10}y^{\frac{1}{3}}$.
According to the power rule of logarithms, we know that $log_{b}M^{p}=plog_{b}M$ (when $b$ and $M$ are positive real numbers, $b\ne1$, and $p$ is any real number).
Therefore, $log_{10}x^{\frac{1}{3}}-log_{10}y^{\frac{1}{3}}=\frac{1}{3}log_{10}x-\frac{1}{3}log_{10}y$.
