2. Solve the inequality $(\sqrt[10]{125})^{\log _{\sqrt{5}}^{2} x}+3 \geqslant x^{\log _{5} x}+3(\sqrt[5]{x})^{\log _{5} x}$.
Answer. $x \in\left(0 ; 5^{-\sqrt{\log _{5} 3}}\right] \cup\{1\} \cup\left[5 \sqrt{\log _{5} 3} ;+\infty\right)$.

Solution. Rewrite the inequality as $5^{\frac{6}{5} \log _{5}^{2} x}+3-5^{\log _{5}^{2} x}-3 \cdot 5^{\frac{1}{5} \log _{5}^{2} x} \geqslant 0$. By grouping the first term with the third and the second with the fourth, we factor the left side of the inequality:

$$
\left(5^{\log _{5}^{2} x}-3\right)\left(5^{\frac{1}{5} \log _{5}^{2} x}-1\right) \geqslant 0
$$

There are two possible cases.
a) $\left\{\begin{array}{l}5^{\log _{5}^{2} x}-3 \geqslant 0, \\ 5^{\frac{1}{5} \log _{5}^{2} x}-1 \geqslant 0\end{array} \Leftrightarrow\left\{\begin{array}{l}\log _{5}^{2} x \geqslant \log _{5} 3, \\ \log _{5}^{2} x \geqslant 0\end{array} \Leftrightarrow\left[\begin{array}{l}\log _{5} x \geqslant \sqrt{\log _{5} 3}, \\ \log _{5} x \leqslant-\sqrt{\log _{5} 3}\end{array} \Leftrightarrow\left[\begin{array}{l}x \geqslant 5^{\sqrt{\log _{5} 3}}, \\ x \leqslant 5^{-\sqrt{\log _{5} 3}} ;\end{array}\right.\right.\right.\right.$

b) $\left\{\begin{array}{l}5^{\log _{5}^{2} x}-3 \leqslant 0, \\ 5^{\frac{1}{5} \log _{5}^{2} x}-1 \leqslant 0\end{array} \Leftrightarrow\left\{\begin{array}{l}\log _{5}^{2} x \leqslant \log _{5} 3, \\ \log _{5}^{2} x \leqslant 0\end{array} \Leftrightarrow \log _{5} x=0 \Leftrightarrow x=1\right.\right.$.

Combining the results, we get $x \in\left(0 ; 5^{-\sqrt{\log _{5} 3}}\right] \cup\{1\} \cup\left[5 \sqrt{\log _{5} 3} ;+\infty\right)$.