Answer
$\displaystyle \frac{2}{3}\log_{5}x+\frac{1}{3}\log_{5}y-\frac{2}{3}$
Work Step by Step
$\log_{5} \displaystyle \sqrt[3]{\frac{x^{2}y}{25}}=\log_{5}\left( \frac{x^{2}y}{25}\right)^{1/3}$
$\quad $...apply the Power Rule: $\quad \log_{b}(M^{p})=p\cdot\log_{b}\mathrm{M}$
$=\displaystyle \frac{1}{3}\left(\log_{5} \frac{x^{2}y}{25}\right)$
$\quad $..apply the Quotient Rule: $\displaystyle \quad \log_{b}(\frac{M}{N})=\log_{b}\mathrm{M}-\log_{b}\mathrm{N}$
$=\displaystyle \frac{1}{3}\left(\log_{5}(x^{2}y)- \log_{5}25\right)$
$\quad $...apply the Product Rule: $\quad \log_{b}(MN)=\log_{b}\mathrm{M}+\log_{b}\mathrm{N}$
$=\displaystyle \frac{1}{3}\left(\log_{5}(x^{2})+\log_{5}y- \log_{5}5^{2}\right)$
$\quad $...apply the Power Rule: $\quad \log_{b}(M^{p})=p\cdot\log_{b}\mathrm{M}$
$\quad $... also, $\log_{b}b^{x}=x\ \Rightarrow\ \log_{5}5^{2}=2$
$=\displaystyle \frac{1}{3}\left(2\log_{5}x+\log_{5}y- 2\right)\quad $... distribute
$=\displaystyle \frac{2}{3}\log_{5}x+\frac{1}{3}\log_{5}y-\frac{2}{3}$
