Example 18. Solve the inequality

$$
5^{\frac{1}{4} \log _{5}^{2} x} \geqslant 5 x^{\frac{1}{5} \log _{5} x}
$$
Solution. The domain of admissible values of inequality (14) is defined by the condition $x>0$. For such $x$, both sides of inequality (14) are positive. Taking the logarithm of both sides to the base 5, we obtain the inequality

$$
\frac{1}{4} \log _{5}^{2} x \geqslant 1+\frac{1}{5} \log _{5}^{2} x
$$

equivalent to (14). From (15), we get the inequality

$$
\log _{5}^{2} x \geqslant 20
$$

i.e.

$$
\left|\log _{5} x\right| \geqslant 2 \sqrt{5}
$$

Thus, inequality (14) is equivalent to the system

$$
\left[\begin{array}{l}
\operatorname{log}_{5} x \geqslant 2 \sqrt{5} \\
\operatorname{log}_{5} x \leqslant -2 \sqrt{5}
\end{array} \Leftrightarrow \left[\begin{array}{l}
x \geqslant 5^{2} \sqrt{5} \\
0<x \leqslant 5^{-2} \sqrt{5}
\end{array}\right.\right.
$$

Therefore, the solution set of inequality (14) is $\left(0 ; 5^{-2 \sqrt{5}}\right] \cup\left[5^{2} \sqrt{5} ;+\infty\right)$.