3・1 Simplify:
$$\log _{a}\left[\left(\frac{m^{4} n^{-4}}{m^{-1} n}\right)^{-3} \div\left(\frac{m^{-2} n^{2}}{m n^{-1}}\right)^{5}\right]$$

Here $m$, $n$, and $a$ are all positive numbers, $a \neq 1$.
[Solution] $\begin{aligned} & \log _{a}\left[\left(\frac{m^{4} n^{-4}}{m^{-1} n}\right)^{-3} \div\left(\frac{m^{-2} n^{2}}{m n^{-1}}\right)^{5}\right] \\ = & \log _{a}\left[\left(m^{5} n^{-5}\right)^{-3} \div\left(m^{-3} n^{3}\right)^{5}\right] \\ = & \log _{a}\left(m^{-15} n^{15} \div m^{-15} n^{15}\right) \\ = & \log _{a} 1 \\ = & 0\end{aligned}$