Answer
$\log_a \left(\dfrac{x^{\frac{2}{3}}z^5}{y^{\frac{1}{2}}}\right)$
Work Step by Step
Using the properties of logarithms, the given expression, $
\dfrac{2}{3}\log_a x-\dfrac{1}{2}\log_a y+5\log_a z
,$ is equivalent to
\begin{align*}
&
\log_a x^{\frac{2}{3}}-\log_a y^{\frac{1}{2}}+\log_a z^5
&(\log_b x^y=y\log_b x)
\\\\&=
\log_a x^{\frac{2}{3}}+\log_a z^5-\log_a y^{\frac{1}{2}}
\\\\&=
\log_a \left(x^{\frac{2}{3}}z^5\right)-\log_a y^{\frac{1}{2}}
&(\log_b (xy)=\log_b x+\log_b y)
\\\\&=
\log_a \left(\dfrac{x^{\frac{2}{3}}z^5}{y^{\frac{1}{2}}}\right)
&(\log_b \dfrac{x}{y}=\log_b x-\log_b y)
.\end{align*}
Hence, as a single logarithm, the given expression is equivalent to $\log_a \left(\dfrac{x^{\frac{2}{3}}z^5}{y^{\frac{1}{2}}}\right)$.
