Question

The value of $x$ satisfying the equation $g^{lo{g}_3\left(lo{g}_{2}x\right)}=lo{g}_{2}x-(lo{g}_{2}x{)}^2+1$ is

• A. $0$
• B. $1$
• C. $2$
• D. None

Answer 1

The correct option is C $2$

Here, $g^{{\log }_3\left({\log }_{2}x\right)}$ is an exponential function and ${\log }_{2}x-{\left({\log }_{2}x\right)}^2+1$ is a quadratic with imaginary roots.

The two can be equal when both side become $0,1$. Since, right hand side can become zero at imaginary point. We, only consider, then the two side become $1$.

${\log }_{2}x-{\left({\log }_{2}x\right)}^2+1=1$

$\Rightarrow {\left({\log }_{2}x\right)}^2-\left({\log }_{2}x\right)=0$

$\Rightarrow \left({\log }_{2}x\right)({\log }_{2}x-1)=0$

$\Rightarrow {\log }_{2}x=0$ and ${\log }_{2}x=1$

$\Rightarrow x=1,2$

But $x\ne 1$ as in $g^{{\log }_3\left({\log }_{2}x\right)}$ it become invalid hence, $x=2$ satisfy the relation.