Answer
$2$
Work Step by Step
Using the properties of logarithms, the given expression, $
\log_a \dfrac{\sqrt[3]{x^2z}}{\sqrt[3]{y^2z^{-1}}},$ is equivalent to
\begin{align*}\require{cancel}
&
\log_a \sqrt[3]{x^2z}-\log_a\sqrt[3]{y^2z^{-1}}
&(\log_a\dfrac{x}{y}=\log_a x-\log_ay)
\\\\&=
\log_a (x^2z)^{1/3}-\log_a(y^2z^{-1})^{1/3}
\\\\&=
\dfrac{1}{3}\log_a (x^2z)-\dfrac{1}{3}\log_a(y^2z^{-1})
&(\log_b x^y=y\log_b x)
\\\\&=
\dfrac{1}{3}(\log_a x^2+\log_a z)-\dfrac{1}{3}(\log_a y^2+\log_az^{-1})
&(\log_b xy=\log_b x+\log_b y)
\\\\&=
\dfrac{1}{3}(2\log_a x+\log_a z)-\dfrac{1}{3}(2\log_a y-1\log_az)
&(\log_b x^y=y\log_b x)
\\\\&=
\dfrac{2}{3}\log_a x+\dfrac{1}{3}\log_a z-\dfrac{2}{3}\log_a y+\dfrac{1}{3}\log_az
\\\\&=
\dfrac{2}{3}\log_a x-\dfrac{2}{3}\log_a y+\left(\dfrac{1}{3}\log_a z+\dfrac{1}{3}\log_az\right)
\\\\&=
\dfrac{2}{3}\log_a x-\dfrac{2}{3}\log_a y+\dfrac{2}{3}\log_a z
.\end{align*}
Substituting the given values, $\log_a x=2,$ $\log_ay=3,$ and $\log_a z=4,$ the expression above is equivalent to
\begin{align*}
&
\dfrac{2}{3}(2)-\dfrac{2}{3}(3)+\dfrac{2}{3}(4)
\\\\&=
\dfrac{4}{3}-\dfrac{2}{\cancel3}(\cancel3)+\dfrac{8}{3}
\\\\&=
\dfrac{4}{3}-2+\dfrac{8}{3}
\\\\&=
\left(\dfrac{4}{3}+\dfrac{8}{3}\right)-2
\\\\&=
\dfrac{12}{3}-2
\\\\&=
4-2
\\\\&=
2
.\end{align*}
Hence, the given expression evaluates to $2$.
