1. Solve the inequality

$$
\log _{\frac{1}{3}}\left(\log _{4}\left(x^{2}-5\right)\right)>0
$$
Solution. Since

$$
\log _{\frac{1}{3}} x=\log _{3} x^{-1}=-\log _{3} x
$$

we get

$$
-\log _{3}\left(\log _{4}\left(x^{2}-5\right)\right)>0
$$

from which it follows that $0<\log _{4}\left(x^{2}-5\right)<1$. Therefore, $1<x^{2}-5<4, \quad$ i.e., $6<x^{2}<9$. The solution to the inequality is $(-3,-\sqrt{6}) \cup(\sqrt{6}, 3)$.