2. Solve the inequality $x^{\log _{13} x}+7(\sqrt[3]{x})^{\log _{13} x} \leqslant 7+(\sqrt[3]{13})^{\log _{\sqrt{13}}^{2} x}$
Answer. $x \in\left(0 ; 13^{-\sqrt{\log _{13} 7}}\right] \cup\{1\} \cup\left[13 \sqrt{\log _{13} 7} ;+\infty\right)$.

Solution. Rewrite the inequality as $13^{\log _{13}^{2} x}+7 \cdot 13^{\frac{1}{3} \log _{13}^{2} x}-7-13^{\frac{4}{3} \log _{13}^{2} x} \leqslant 0$. By grouping the first term with the fourth and the second with the third, we factor the left side of the inequality:

$$
\left(13^{\log _{13}^{2} x}-7\right)\left(1-13^{\frac{1}{3} \log _{13}^{2} x}\right) \leqslant 0
$$

There are two possible cases.
a) $\left\{\begin{array}{l}13^{\log _{13}^{2} x}-7 \geqslant 0, \\ 1-13^{\frac{1}{3} \log _{13}^{2} x} \leqslant 0\end{array} \Leftrightarrow\left\{\begin{array}{l}\log _{13}^{2} x \geqslant \log _{13} 7, \\ \log _{13}^{2} x \geqslant 0\end{array} \Leftrightarrow\left[\begin{array}{l}\log _{13} x \geqslant \sqrt{\log _{13} 7}, \\ \log _{13} x \leqslant-\sqrt{\log _{13} 7}\end{array} \Leftrightarrow\left[\begin{array}{l}x \geqslant 13^{\sqrt{\log _{13} 7}}, \\ x \leqslant 13^{-\sqrt{\log _{13} 7}} ;\end{array}\right.\right.\right.\right.$
b) $\left\{\begin{array}{l}13^{\log _{13}^{2} x}-7 \leqslant 0, \\ 1-13^{\frac{1}{3} \log _{13}^{2} x} \geqslant 0\end{array} \Leftrightarrow\left\{\begin{array}{l}\log _{13}^{2} x \leqslant \log _{13} 7, \\ \log _{13}^{2} x \leqslant 0\end{array} \Leftrightarrow \log _{13} x=0 \Leftrightarrow x=1\right.\right.$.
Combining the results, we get $x \in\left(0 ; 13^{-\sqrt{\log _{13} 7}}\right] \cup\{1\} \cup\left[13^{\sqrt{\log _{13} 7}} ;+\infty\right)$.