2. Solve the equation

$$
3^{\log _{x} 3} \cdot x^{\log _{3} x}=9
$$
Solution. We take the logarithm of the equation with base 3, from which we obtain the equation

$$
\log _{x} 3+\left(\log _{3} x\right)^{2}=2
$$

Setting $\log _{3} x=u$ and using the identity

$$
\log _{a} b=\frac{1}{\log _{b} a}
$$

we get the equation $\frac{1}{u}+u^{2}=2$, i.e., $u^{3}-2 u+1=0$. This equation has a solution $u_{1}=1$, and using the method as in the previous problem, the other two solutions are $u_{2}=\frac{-1+\sqrt{5}}{2}$ and $u_{3}=\frac{-1-\sqrt{5}}{2}$. Therefore, the solutions to the given equation are $x_{1}=3$, $x_{2}=3^{\frac{-1+\sqrt{5}}{2}}$, and $x_{3}=3^{\frac{-1-\sqrt{5}}{2}}$.