1. Solve the equation $x^{\log _{2}(8 x)}=\frac{x^{7}}{8}$.
Answer: $x=2, x=8$.

Solution. By logarithmizing with base 2, we get $\log _{2} x \cdot \log _{2}(8 x)=\log _{2} x^{7}-\log _{2} 8$, which is equivalent to the following: $\log _{2}^{2} x+3 \log _{2} x=7 \log _{2} x-3 \Leftrightarrow \log _{2}^{2} x-4 \log _{2} x+3=0$, from which $\log _{2} x=1$ or $\log _{2} x=3 ; x=2$ or $x=8$.