1. Solve the equation $x^{\log _{3}\left(27 x^{2}\right)}=\frac{x^{9}}{81}$.
Answer: $x=3, x=9$.

Solution. By taking the logarithm with base 3, we get $\log _{3} x \cdot \log _{3}\left(27 x^{2}\right)=\log _{3} x^{9}-\log _{3} 81$, which is equivalent to the following: $2 \log _{3}^{2} x+3 \log _{3} x=9 \log _{3} x-4 \Leftrightarrow \log _{3}^{2} x-3 \log _{3} x+2=0$, from which $\log _{3} x=1$ or $\log _{3} x=2 ; x=3$ or $x=9$.