Answer
$\dfrac{1}{2}\log_5 x-\log_5 y-\log_5 z$
Work Step by Step
Using the properties of logarithms, the given expression, $
\log_5 \left(\dfrac{\sqrt{x}}{yz}\right)
,$ is equivalent to
\begin{align*}\require{cancel}
&
\log_5 \sqrt{x}-\log_5 (yz)
&(\text{use }\log_b \dfrac{x}{y}=\log_b x-\log_b y)
\\&=
\log_5 \sqrt{x}-\left(\log_5 y+\log_5 z\right)
&(\text{use }\log_b (xy)=\log_b x+\log_b y)
\\&=
\log_5 x^{1/2}-\log_5 y-\log_5 z
\\&=
\dfrac{1}{2}\log_5 x-\log_5 y-\log_5 z
&(\text{use }\log_b x^y=y\log_b x)
.\end{align*}
Hence, the expression $
\log_5 \left(\dfrac{\sqrt{x}}{yz}\right)
$ is equivalent to $
\dfrac{1}{2}\log_5 x-\log_5 y-\log_5 z
$.
