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

The correct option is D $3\sqrt{3}$

Given,

${\log }_2\left({\log }_{8}x\right)={\log }_8\left({\log }_{2}x\right)$

or, ${\log }_2\left({\log }_{2^3}x\right)={\log }_{2^3}\left({\log }_{2}x\right)$

or, ${\log }_2\frac{1}{3}\left({\log }_{2}x\right)=\frac{1}{3}{\log }_2\left({\log }_{2}x\right)$

or, ${\log }_2\frac{1}{3}\left({\log }_{2}x\right)={\log }_2{\left({\log }_{2}x\right)}^{\frac{1}{3}}$

or, $\frac{1}{3}{\log }_{2}x=({\log }_{2}x{)}^{\frac{1}{3}}$

or, $\frac{1}{3^3}({\log }_{2}x{)}^3=({\log }_{2}x)$

or, $({\log }_{2}x{)}^2=27$

or, ${\log }_{2}x=\sqrt{27}=3\sqrt{3}$.