Answer
$\dfrac{7}{2}\log_a x-\dfrac{5}{2}\log_a y-4\log_az$
Work Step by Step
Using the properties of logarithms, the given expression, $
\log_a \sqrt{\dfrac{x^7}{y^5z^8}}
,$ is equivalent to
\begin{array}{l}\require{cancel}
\log_a \left( \dfrac{x^7}{y^5z^8} \right)^{1/2}
\\\\=
\dfrac{1}{2}\log_a \dfrac{x^7}{y^5z^8}
\\\\=
\dfrac{1}{2}\left[ \log_a x^7-\log_a (y^5z^8) \right]
\\\\=
\dfrac{1}{2}\left[ \log_a x^7-\left( \log_a y^5+\log_az^8 \right) \right]
\\\\=
\dfrac{1}{2}\left[ \log_a x^7-\log_a y^5-\log_az^8 \right]
\\\\=
\dfrac{1}{2}\left[ 7\log_a x-5\log_a y-8\log_az \right]
\\\\=
\dfrac{1}{2}\cdot7\log_a x-\dfrac{1}{2}\cdot5\log_a y-\dfrac{1}{2}\cdot8\log_az
\\\\=
\dfrac{7}{2}\log_a x-\dfrac{5}{2}\log_a y-\dfrac{8}{2}\log_az
\\\\=
\dfrac{7}{2}\log_a x-\dfrac{5}{2}\log_a y-4\log_az
.\end{array}
