Answer
$2\log_a x+3\log_a y-\dfrac{3}{4}\log_a a-\dfrac{5}{4}\log_az$
Work Step by Step
Using the properties of logarithms, the given expression, $
\log_a \sqrt[4]{\dfrac{x^8y^{12}}{a^3z^5}}
,$ is equivalent to
\begin{array}{l}\require{cancel}
\log_a \left(\dfrac{x^8y^{12}}{a^3z^5} \right)^{1/4}
\\\\=
\dfrac{1}{4}\log_a \left(\dfrac{x^8y^{12}}{a^3z^5} \right)
\\\\=
\dfrac{1}{4}\left[ \log_a \left(x^8y^{12}\right)-\log_a \left(a^3z^5 \right) \right]
\\\\=
\dfrac{1}{4}\left[ \log_a x^8+ \log_a y^{12}-\left(\log_a a^3+\log_az^5 \right) \right]
\\\\=
\dfrac{1}{4}\left[ \log_a x^8+ \log_a y^{12}-\log_a a^3-\log_az^5 \right]
\\\\=
\dfrac{1}{4}\left[ 8\log_a x+ 12\log_a y-3\log_a a-5\log_az \right]
\\\\=
\dfrac{1}{4}(8\log_a x)+\dfrac{1}{4}( 12\log_a y)-\dfrac{1}{4}(3\log_a a)-\dfrac{1}{4}(5\log_az)
\\\\=
\dfrac{8}{4}\log_a x+\dfrac{12}{4}\log_a y-\dfrac{3}{4}\log_a a-\dfrac{5}{4}\log_az
\\\\=
2\log_a x+3\log_a y-\dfrac{3}{4}-\dfrac{5}{4}\log_az
.\end{array}
