Track / Stage 4 / 122 of 340 #382 of 1964

Problem 382

AMC 12 late, AIME early

Algebra Difficulty 4.7 Find the answer

$\log _{3} \log _{x^{2}} \log _{x^{2}} x^{4}>0$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

## Solution.

The given inequality is equivalent to the inequality $\log _{x^{2}} \log _{x^{2}}\left(x^{2}\right)^{2}>1 \Leftrightarrow$ $\Leftrightarrow \log _{x^{2}} 2 \log _{x^{2}} x^{2}>1, \log _{x^{2}} 2>1 \Leftrightarrow$

Answer: $x \in(-\sqrt{2} ;-1) \cup(1 ; \sqrt{2})$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.