Question

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

• A. 9
• B. 12
• C. 27
• D. $3\sqrt{3}$

Answer 1

Answer: (C)

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