1. Solve the equation $x^{\log _{2}\left(0.25 x^{3}\right)}=512 x^{4}$.
Answer: $x=\frac{1}{2}, x=8$.

Solution. By logarithmizing with base 2, we get $\log _{2} x \cdot \log _{2}\left(\frac{1}{4} x^{3}\right)=\log _{2} x^{4}+\log _{2} 512$, which is equivalent to the following: $3 \log _{2}^{2} x-2 \log _{2} x=4 \log _{2} x+9 \Leftrightarrow \log _{2}^{2} x-2 \log _{2} x-3=0$, from which $\log _{2} x=-1$ or $\log _{2} x=3$; $x=\frac{1}{2}$ or $x=8$.