Answer
$\displaystyle \frac{1}{2}\log_{4}x-3$
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_{4}\frac{\sqrt{x}}{64}\qquad $... apply Quotient Rule
$=\log_{4}\sqrt{x}-\log_{4}64\qquad $...$\sqrt{x}=x^{1/2}, 64=4^{3}$
$=\log_{4}x^{1/2}-\log_{4}4^{3}\qquad $... apply Power Rule
$=\displaystyle \frac{1}{2}\log_{4}x-\log_{4}4^{3}\qquad $... apply $\log_{b}b^{x}=x $
=$\displaystyle \frac{1}{2}\log_{4}x-3$
