4. Solve the system

$$
\left\{\begin{array}{l}
\log _{y} x-2 \log _{x} y=1 \\
x-y=2
\end{array}\right.
$$
Solution. Let $t=\log _{y} x$; then the first equation reads $t-\frac{2}{t}=1$, from which we get $t_{1}=2$ and $t_{2}=-1$.

From $\log _{y} x=2$, we get $x=y^{2}$. From the second equation, $x-y=2$ and $x=y^{2}$, it follows that $y=2$, since $y=-1$ is not taken as a solution. Thus, one solution is $x=4$ and $y=2$.

From $\log _{y} x=-1$ we get $x=\frac{1}{y}$. From this and the second equation, it follows that $y=\sqrt{2}-1(y>0)$. Thus, the second solution is $x=\sqrt{2}+1$ and $y=\sqrt{2}-1$.

## 4th year