Problem 785

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 $\left(0 ; 5^{-2 \sqrt{5}}\right] \cup\left[5^{2} \sqrt{5} ;+\infty\right)$.