1. Solve the equation

$$
5^{x} \sqrt[x]{8^{x-1}}=500
$$
Solution. By exponentiation of the given equation, we get

$$
5^{x^{2}} \cdot 8^{x-1}=2^{2 x} \cdot 5^{3 x}, \text { i.e. } 5^{x^{2}-3 x}=2^{3 x}
$$

Further, by taking the logarithm, we will have

$$
\begin{aligned}
& \left(x^{2}-3 x\right) \log _{2} 5=3-x \\
& (x-3)\left(x \log _{2} 5+1\right)=0
\end{aligned}
$$

from which we get $x_{1}=3, x_{2}=-\log _{5} 2$.