1. Solve the equation $x^{\log _{5}(0.008 x)}=\frac{125}{x^{5}}$.
Answer: $x=5, x=\frac{1}{125}$.

Solution. By logarithmizing with base 5, we get $\log _{5} x \cdot \log _{5}\left(\frac{x}{125}\right)=\log _{5} 125-\log _{5} x^{3}$, which is equivalent to the following: $\log _{5}^{2} x-3 \log _{5} x=3-5 \log _{5} x \Leftrightarrow \log _{5}^{2} x+2 \log _{5} x-3=0$, from which $\log _{5} x=1$ or $\log _{5} x=-3 ; x=5$ or $x=\frac{1}{125}$.