Answer
$\dfrac{1}{4}\log_4 z+\dfrac{1}{5}\log_4 w-2\log_4 s$
Work Step by Step
Using the properties of logarithms, the given expression, $
\log_4\dfrac{\sqrt[4]{z}\cdot\sqrt[5]{w}}{s^2}
$, is equivalent to
\begin{align*}
&
\log_4\left(\sqrt[4]{z}\cdot\sqrt[5]{w}\right)-\log_4s^2
\\\\&=
\log_4\sqrt[4]{z}+\log_4\sqrt[5]{w}-\log_4s^2
&(\text{use }\log_b (xy)=\log_b x+\log_b y)
\\\\&=
\log_4 z^{1/4}+\log_4 w^{1/5} -\log_4s^2
\\\\&=
\dfrac{1}{4}\log_4 z+\dfrac{1}{5}\log_4 w-2\log_4 s
&(\text{use }\log_b x^y=y\log_b x)
.\end{align*}
Hence, the expression $
\log_4\dfrac{\sqrt[4]{z}\cdot\sqrt[5]{w}}{s^2}
$ is equivalent to $
\dfrac{1}{4}\log_4 z+\dfrac{1}{5}\log_4 w-2\log_4 s
$.
