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Problem 346

AMC 12 late, AIME early

Algebra Difficulty 4.7 Find the answer

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

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 and solution reproduced as published; topic, difficulty and ordering added by this site.