Answer
$\dfrac{1}{2}\log_3 x +\dfrac{1}{2}\log_3 y-\dfrac{1}{2}\log_3 5$
Work Step by Step
Using the properties of logarithms, the given expression, $
\log_3\sqrt{\dfrac{xy}{5}}
$, is equivalent to
\begin{align*}
&
\log_3\left(\dfrac{xy}{5}\right)^{1/2}
\\\\&=
\dfrac{1}{2}\log_3 \dfrac{xy}{5}
&(\text{use }\log_b x^y=y\log_b x)
\\\\&=
\dfrac{1}{2}\left(\log_3 (xy)-\log_3 5\right)
&(\text{use }\log_b \dfrac{x}{y}=\log_b x-\log_b y)
\\\\&=
\dfrac{1}{2}\left(\log_3 x +\log_3 y-\log_3 5\right)
&(\text{use }\log_b (xy)=\log_b x+\log_b y)
\\\\&=
\dfrac{1}{2}(\log_3 x) +\dfrac{1}{2}(\log_3 y)-\dfrac{1}{2}(\log_3 5)
&(\text{use Distributive Property}
\\\\&=
\dfrac{1}{2}\log_3 x +\dfrac{1}{2}\log_3 y-\dfrac{1}{2}\log_3 5
.\end{align*}
Hence, the expression $
\log_3\sqrt{\dfrac{xy}{5}}
$ is equivalent to $
\dfrac{1}{2}\log_3 x +\dfrac{1}{2}\log_3 y-\dfrac{1}{2}\log_3 5
$.
