Question

The number of values of $x$ satisfying the equation $lo{g}_2\left(lo{g}_3\right(lo{g}_{2}x\left)\right)\ge \sqrt{(x-8)}+\sqrt{(8-x)}$ is

• A. $Zero$
• B. $1$
• C. $3$
• D. $4$

Answer 1

The correct option is B $1$

${\log }_2\left({\log }_3\right({\log }_{2}x\left)\right)\ge \sqrt{x-8}+\sqrt{8-x}$

For right hand side to be defined.

$x-8\ge 0$ and $8-x\ge 0$

$\Rightarrow x=8$ is the only value.

for which R.H.S is defined.

$\Rightarrow$ at $x=8$

L.H.S$={\log }_2\left({\log }_3\right({\log }_{2}8\left)\right)={\log }_2\left({\log }_3\right(3{\log }_{2}2\left)\right)$

$={\log }_2\left({\log }_{3}3\right)={\log }_{2}1=0$

R.H.S $=\sqrt{x-8}+\sqrt{8-x}$

at $x=8$

R.H.S$=0$

$\Rightarrow$ L.H.S $=$ R.H.S at $x=8$

$\Rightarrow x=8$ is the only solution.