Answer
$=\displaystyle \frac{10}{3}$
Work Step by Step
... apply $ \displaystyle \log_{a}\frac{M}{N}=\log_{a}M-\log_{a}N$
$\displaystyle \log_{a}\frac{\sqrt[3]{x^{2}z}}{\sqrt[3]{y^{2}z^{-2}}} =\log_{a}\sqrt[3]{x^{2}z}-\log_{a}\sqrt[3]{y^{2}z^{-2}}$
$...$ rewrite $\sqrt[3]{A}$ as $A^{1/3}$
$=\log_{a}(x^{2}z)^{1/3}-\log_{a}(y^{2}z^{-2})^{1/3}$
... apply $\log_{a}M^{p}=p\cdot\log_{a}M$
$=\displaystyle \frac{1}{3}\log_{a}(x^{2}z) -\frac{1}{3}\log_{a}(y^{2}z^{-2})^{ }$
... apply $\log_{a}(MN)=\log_{a}M+\log_{a}N$
$=\displaystyle \frac{1}{3}\log_{a}(x^{2})+\frac{1}{3}\log_{a}(z) -\frac{1}{3}\log_{a}(y^{2})^{ }-\frac{1}{3}\log_{a}(z^{-2})^{ }$
... apply $\log_{a}M^{p}=p\cdot\log_{a}M$
$=\displaystyle \frac{2}{3}\log_{a}x+\frac{1}{3}\log_{a}z -\frac{2}{3}\log_{a}y-\frac{-2}{3}\log_{a}z$
$=\displaystyle \frac{2}{3}\log_{a}x-\frac{2}{3}\log_{a}y+\frac{1}{3}\log_{a}z+\frac{2}{3}\log_{a}z$
$=\displaystyle \frac{2}{3}\log_{a}x-\frac{2}{3}\log_{a}y+\log_{a}z$
Given $\log_{a}x=2, \log_{a}y=3$, and $\log_{a}z=4$
$=\displaystyle \frac{2}{3}(2)-\frac{2}{3}(3)+4$
$=\displaystyle \frac{4-6+12}{3}$
$=\displaystyle \frac{10}{3}$
