Question

Find all real numbers x which satisfy the equation $2{\log }_2{\log }_{2}x+{\log }_{1/2}{\log }_2\left(2\sqrt{2}x\right)=1$.

Answer 1

Given $2{\log }_2{\log }_{2}x+{\log }_{1/2}{\log }_2\left(2\sqrt{2}x\right)=1$

Let ${\log }_{2}x=y$

$\Rightarrow 2{\log }_{2}y+{\log }_{2^{-1}}({\log }_2{2}^{3/2}+y)=1$

$\Rightarrow {\log }_2{y}^2-{\log }_2(y+\frac{3}{2})=1$

$\Rightarrow \frac{y^2}{y+\frac{3}{2}}=2$

$\Rightarrow y^2-2y-3=0$

$\Rightarrow y=3,-1$

$\Rightarrow {\log }_{2}x=3,-1$

$\Rightarrow x=8,\frac{1}{2}$