Answer
$\displaystyle \frac{1}{5}\log_{2}x+\frac{4}{5}\log_{2}y-\frac{4}{5}$
Work Step by Step
$\displaystyle \log_{2}\sqrt[5]{\frac{xy^{4}}{16}}=\log_{2}\left( \frac{xy^{4}}{16}\right)^{1/5}$
$\quad $...apply the Power Rule: $\quad \log_{b}(M^{p})=p\cdot\log_{b}\mathrm{M}$
$=\displaystyle \frac{1}{5}\left(\log_{2} \frac{xy^{4}}{16}\right)$
$\quad $..apply the Quotient Rule: $\displaystyle \quad \log_{b}(\frac{M}{N})=\log_{b}\mathrm{M}-\log_{b}\mathrm{N}$
$=\displaystyle \frac{1}{5}\left(\log_{2}(xy^{4})- \log_{2}16\right)$
$\quad $...apply the Product Rule: $\quad \log_{b}(MN)=\log_{b}\mathrm{M}+\log_{b}\mathrm{N}$
$=\displaystyle \frac{1}{5}\left(\log_{2}x+\log_{2}y^{4}- \log_{5}2^{4}\right)$
$\quad $...apply the Power Rule: $\quad \log_{b}(M^{p})=p\cdot\log_{b}\mathrm{M}$
$\quad $... also, $\log_{b}b^{x}=x\ \Rightarrow\ \log_{2}2^{4}=4$
$=\displaystyle \frac{1}{5}\left(\log_{2}x+4\log_{2}y- 4\right)\quad $... distribute
$=\displaystyle \frac{1}{5}\log_{2}x+\frac{4}{5}\log_{2}y-\frac{4}{5}$
