Question

Find the number of values of $x$ satisfying the equation $2{\left({\log }_{2}x\right)}^2+(1-\sqrt{2}){\log }_2{x}^2=2\sqrt{2}$ is

Answer 1

Clearly $x>0$.
$2{\left({\log }_{2}x\right)}^2+(1-\sqrt{2}){\log }_2{x}^2=2\sqrt{2}$

$\Rightarrow 2{\left({\log }_{2}x\right)}^2+2(1-\sqrt{2}){\log }_{2}x=2\sqrt{2}[\because \log a^m=m\log a]$

Put ${\log }_{2}x=t$ to obtain

$2t^2+2(1-\sqrt{2})t-2\sqrt{2}=0$

$\Rightarrow (t+1)(t-\sqrt{2})=0$

$\Rightarrow x=\frac{1}{2},2^{\sqrt{2}}$
Ans: 2