Answer
$\dfrac{1}{2}\log_4 x+2\log_4 w-\log_4z$
Work Step by Step
Using the properties of logarithms, the given expression, $
\log_4\dfrac{\sqrt{x}\cdot w^2}{z}
$, is equivalent to
\begin{align*}\require{cancel}
&
\log_4 \left(\sqrt{x}\cdot w^2\right)-\log_4z
&(\text{use }\log_b \dfrac{x}{y}=\log_b x-\log_b y)
\\\\&=
\log_4 \sqrt{x}+\log_4 w^2-\log_4z
&(\text{use }\log_b (xy)=\log_b x+\log_b y)
\\\\&=
\log_4 x^{1/2}+\log_4 w^2-\log_4z
\\\\&=
\dfrac{1}{2}\log_4 x+2\log_4 w-\log_4z
&(\text{use }\log_b x^y=y\log_b x)
.\end{align*}
Hence, the expression $
\log_4\dfrac{\sqrt{x}\cdot w^2}{z}
$ is equivalent to $
\dfrac{1}{2}\log_4 x+2\log_4 w-\log_4z
$.
