log-2-log-3-log-2-2-x-1-lt-1-has-solution-1-a-26-lt-x-lt-b-find-a-

Question Number 81364 by jagoll last updated on 12/Feb/20

\log_{2} (\log_{3} (\log_{2} \frac{2}{x}) - 1) < 1

has solution \frac{1}{a^{26}} < x < b . find a ?

Commented by john santu last updated on 12/Feb/20

(i) \log_{3} (\log_{2} \frac{2}{x}) - 1 > 0

(ii) \log_{3} (\log_{2} \frac{2}{x}) - 1 < 2

\Rightarrow 0 < \log_{3} (\log_{2} \frac{2}{x}) - 1 < 2

\Rightarrow 1 < \log_{3} (\log_{2} \frac{2}{x}) < 3

\Rightarrow 3 < \log_{2} (\frac{2}{x}) < 27

\Rightarrow 2^{3} < \frac{2}{x} < 2^{27}

\Rightarrow \frac{1}{2^{27}} < \frac{x}{2} < \frac{1}{2^{3}}

\Rightarrow \frac{1}{2^{26}} < x < \frac{1}{2^{2}} . we get a = 2

Commented by jagoll last updated on 12/Feb/20

thank mister

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