Answer
$\log_{2}x+2\log_{2}y-6$
Work Step by Step
Basic logarithmic properties:$ \left\{\begin{array}{l}
\log_{b}1=0\\
\log_{b}b=1\\
\log_{b}b^{x}=x\\
b^{\log_{b}}x=x
\end{array}\right.$
Rules:
The Product Rule: $\log_{b}(MN)=\log_{b}\mathrm{M}+\log_{b}\mathrm{N}$
The Quotient Rule: $\displaystyle \log_{b}(\frac{M}{N})=\log_{b}\mathrm{M}-\log_{b}\mathrm{N}$
The Power Rule: $\log_{b}(M^{p})=p\cdot\log_{b}\mathrm{M}$
---
$\displaystyle \log_{2}\frac{xy^{2}}{64}=\qquad $... apply Quotient Rule
$=\log_{2}(xy^{2})-\log_{2}64=\qquad $... apply Product Rule
$=\log_{2}x+\log_{2}y^{2}-\log_{2}64\qquad $... recognize $64=2^{6}$
$=\log_{2}x+\log_{2}y^{2}-\log_{2}2^{6}\qquad $... apply Power Rule
$=\log_{2}x+2\log_{2}y-6\log_{2}2\qquad $... apply $\log_{b}b=1$
$=\log_{2}x+2\log_{2}y-6\cdot 1$
= $\log_{2}x+2\log_{2}y-6$
