## 1. Solve the system of equations

$$
\begin{aligned}
& x^{m}=y^{n} \\
& \log _{a} \frac{x}{y}=\frac{\log _{a} x}{\log _{a} y .}
\end{aligned}
$$
Solution. The system makes sense if and only if $m, n \in \mathbb{R}$ and $a>0, a \neq 1$. Moreover, the unknowns must satisfy the conditions $x, y>0$ and $x \neq 1$. We introduce the substitution $\log _{a} x=u, \log _{a} y=v$, so the given system takes the form

$$
m u=n v, u-v=\frac{u}{v}
$$

In the case where $m=0$, it must also be $n=0$ (it is not possible $v=0$). Then the system reduces to the equation $u-v=\frac{u}{v}$, whose solutions are pairs of the form $\left(\frac{v^{2}}{v-1}, v\right)$, for $v \neq 1$, so in this case the solutions of the given system are pairs of the form

$$
\left(y^{\frac{\log _{a} y}{\log _{a} y-1}}, y\right), y>0, y \neq 1, y \neq a
$$

Let us assume that $m \neq 0$. Then solving the system (1) we find that for $m=n$ it has no solution, and for $m \neq n$ the solution is the pair $\left(\frac{n^{2}}{m(n-m)}, \frac{n}{n-m}\right)$. Then the solution of the given system is $\left(a^{\frac{n^{2}}{m(n-m)}}, a^{\frac{n}{n-m}}\right)$.