2. Solve the equation $\left(\frac{x}{4}\right)^{\log _{5}(50 x)}=x^{6}$.
Answer: $x=\frac{1}{2}, x=2500$.

Solution. By logarithmizing with base 5, we obtain an equation equivalent to the original:

$$
\log _{5}(50 x) \cdot \log _{5}\left(\frac{x}{4}\right)=\log _{5}\left(x^{6}\right) \Leftrightarrow\left(\log _{5} x+2+\log _{5} 2\right)\left(\log _{5} x-2 \log _{5} 2\right)=6 \log _{5} x
$$

Let $\log _{5} x=y, \log _{5} 2=a$. After expanding the brackets and combining like terms, the equation takes the form $y^{2}-(a+4) y-2 a^{2}-4 a=0$. We solve the quadratic equation for $y$:

$$
D=(a+4)^{2}+4\left(2 a^{2}+4 a\right)=9 a^{2}+24 a+16=(3 a+4)^{2}, y=\frac{4+a \pm(3 a+4)}{2} \Leftrightarrow\left[\begin{array}{l}
y=2 a+4, \\
y=-a .
\end{array}\right.
$$

We find $x$: if $y=-a$, then $\log _{5} x=-\log _{5} 2$ and $x=\frac{1}{2}$; if $y=2 a+4$, then $\log _{5} x=2 \log _{5} 2+4$ and $x=2500$.