2. Solve the system of equations

$$
\left\{\begin{array}{l}
(\sqrt{3})^{x-y}=\left(\frac{1}{3}\right)^{2 y-x} \\
\log _{2}(x+y)+\log _{2}(x-y)=4
\end{array}\right.
$$
Solution. We have:

$$
\begin{gathered}
\left\{\begin{array}{c}
3^{\frac{x-y}{2}}=3^{x-2 y} \\
\log _{2}\left(x^{2}-y^{2}\right)=\log _{2} 16
\end{array}\right. \\
\left\{\begin{array}{l}
\frac{x-y}{2}=2 y-x \\
x^{2}-y^{2}=16
\end{array}\right. \\
\left\{\begin{array}{l}
3 x=5 y \\
x^{2}-y^{2}=16
\end{array}\right.
\end{gathered}
$$

from which we obtain $x= \pm 5, y= \pm 3$.