Answer
$ \displaystyle \frac{1}{3}[6\log_{a}x+3\log_{a}y-2-7\log_{a}z]$
Work Step by Step
$\displaystyle \log_{a}\sqrt[3]{\frac{x^{6}y^{3}}{a^{2}z^{7}}}=\log_{a}(\frac{x^{6}y^{3}}{a^{2}z^{7}})^{1/3}\qquad$ ... apply $\log_{a}M^{p}=p\cdot\log_{a}M$
$=\displaystyle \frac{1}{3}\log_{a}(\frac{x^{6}y^{3}}{a^{2}z^{7}})\qquad$ ... apply $\displaystyle \log_{a}\frac{M}{N}=\log_{a}M-\log_{a}N$
$=\displaystyle \frac{1}{3}[\log_{a}(x^{6}y^{3})-\log_{a}(a^{2}z^{7})]\qquad$ ... apply $\log_{a}(MN)=\log_{a}M+\log_{a}N$
$=\displaystyle \frac{1}{3}[\log_{a}x^{6}+\log_{a}y^{3}-(\log_{a}a^{2}+\log_{a}z^{7})]$
$=\displaystyle \frac{1}{3}[\log_{a}x^{6}+\log_{a}y^{3}-\log_{a}a^{2}-\log_{a}z^{7}]\qquad$
... apply $\log_{a}M^{p}=p\cdot\log_{a}M$
$=\displaystyle \frac{1}{3}[6\log_{a}x+3\log_{a}y-2\log_{a}a-7\log_{a}z]\qquad$ ... apply $\log_{a}a=1$
$=\displaystyle \frac{1}{3}[6\log_{a}x+3\log_{a}y-2-7\log_{a}z]$
