2. Solve the equation $\left(\frac{x}{400}\right)^{\log _{5}\left(\frac{x}{8}\right)}=\frac{1024}{x^{3}}$.
Answer: $x=\frac{8}{5}, x=16$.

Solution. By logarithmizing with base 3, we obtain an equation equivalent to the original:

$$
\log _{5}\left(\frac{x}{400}\right) \cdot \log _{5}\left(\frac{x}{8}\right)=\log _{5}\left(\frac{1024}{x^{3}}\right) \Leftrightarrow\left(\log _{5} x-2-4 \log _{5} 2\right)\left(\log _{5} x-3 \log _{5} 2\right)=10 \log _{5} 2-3 \log _{5} x
$$

Let $\log _{5} x=y, \log _{5} 2=a$. After expanding the brackets and combining like terms, the equation takes the form $y^{2}-(7 a-1) y+12 a^{2}-4 a=0$. We solve the quadratic equation for $y$:

$$
D=(7 a-1)^{2}-4\left(12 a^{2}-4 a\right)=a^{2}+2 a+1=(a+1)^{2}, y=\frac{7 a-1 \pm(a+1)}{2} \Leftrightarrow\left[\begin{array}{l}
y=4 a \\
y=3 a-1
\end{array}\right.
$$

We find $x$: if $y=4 a$, then $\log _{5} x=4 \log _{5} 2$ and $x=16$; if $y=3 a-1$, then $\log _{5} x=3 \log _{5} 2-1$ and $x=\frac{8}{5}$.