Solve the following system of equations:

$$
\begin{aligned}
& \log _{a} x-\log _{b} y=m, \\
& \log _{b} x-\log _{c} y=n, \quad \text { where } b=a^{2}, c=a^{3}
\end{aligned}
$$
Let's rewrite the logarithms with bases $b$ and $c$ in terms of base $a$. Based on the identity

$$
\log _{d} e=\frac{\log _{f} e}{\log _{f} d}
$$

our system of equations can be transformed as follows:

$$
\log _{a} x-\frac{1}{2} \log _{a} y=m, \quad \frac{1}{2} \log _{a} x-\frac{1}{3} \log _{a} y=n
$$

which is a system of linear equations in terms of the occurring logarithms. From this,

$$
\begin{aligned}
\log _{a} x & =4 m-6 n, & \log _{a} y & =6 m-12 n \\
x & =a^{4 m-6 n}, & y & =a^{6 m-12 n}
\end{aligned}
$$

Of course, we assumed that $a$ can be used as a logarithmic base, i.e., $a>0$, but $a \neq 1$; then the same holds for $b$ and $c$ as well, and for any pair of values $(m, n)$ (real numbers), $x$ and $y$ are positive numbers.

Martoni Viktor (Veszprém, L. Lovassy Secondary School)