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  4. Given log 2( log 8 x)= log 8( log 2 x) then ( log 2 x)2 has the value equal to

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Q. Given $\log _2\left(\log _8 x\right)=\log _8\left(\log _2 x\right)$ then $\left(\log _2 x\right)^2$ has the value equal to

Continuity and Differentiability

Solution:

Given $\log _2\left(\frac{1}{3} \log _2 x\right)=\frac{1}{3} \log _2\left(\log _2 x\right)$
let $\log _2 x=3 y$
$\Rightarrow \log _2 y =\frac{1}{3} \log _2 3 y \Rightarrow 3 \log _2 y =\log _2 3+\log _2 y \Rightarrow 2 \log _2 y =\log _2 3 $
$\Rightarrow y ^2=3 \Rightarrow \frac{1}{9}\left(\log _2 x \right)^2=3 \Rightarrow \left(\log _2 x \right)^2=27 $