If-log-4-log-1-2-log-3-x-gt-0-then-x-belongs-to-1-a-then-the-value-of-a-2-is-

Question Number 15097 by Tinkutara last updated on 07/Jun/17

If \log_{4} \log_{\frac{1}{2}} \log_{3} (x) > 0 then x belongs

to (1, a), then the value of a^{2} is ?

Commented by prakash jain last updated on 07/Jun/17

\log_{\frac{1}{2}} y > 1 \Rightarrow y < \frac{1}{2}

\log_{3} x < \frac{1}{2} \Rightarrow x < \sqrt{3}

x \in (1, \sqrt{3}) \Rightarrow a = \sqrt{3} \Rightarrow a^{2} = 3

Commented by Tinkutara last updated on 07/Jun/17

Thanks Sir!

Answered by mrW1 last updated on 07/Jun/17

\log_{4} \log_{\frac{1}{2}} \log_{3} (x) > 0

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

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

\Rightarrow x < 3^{1 / 2} = \sqrt{3}

1 < x < \sqrt{3}

\Rightarrow a = \sqrt{3}

\Rightarrow a^{2} = 3

Commented by mrW1 last updated on 07/Jun/17

the answer is 3

Commented by Tinkutara last updated on 07/Jun/17

Thanks Sir!

Answered by Tinkutara last updated on 08/Jun/17

Also finding the domain :

I : x > 0

II : \log_{3} x > 0 \Rightarrow x > 1

III : \log_{\frac{1}{2}} \log_{3} x > 0

\log_{3} x < 1 \Rightarrow x < 3

x < 3 and x < \sqrt{3} \Rightarrow x < \sqrt{3}

∴ x \in (1, \sqrt{3})

\Rightarrow a^{2} = 3