Question

Find $x$ if ${\log }_2{\log }_{\frac{1}{2}}{\log }_{3}x>0$

Answer 1

Here, $x>0$

and ${\log }_{3}x>0$

$\therefore x>1$

And ${\log }_{1/2}{\log }_{3}x>0$

${\log }_{3}x<\left(1/2\right)$

$\therefore (x<3)$

and ${\log }_2{\log }_{1/2}{\log }_{3}x>0$

$\Rightarrow {\log }_{1/2}{\log }_{3}x>1$

$\Rightarrow {\log }_{3}x<\left(1/2\right)$

$\therefore x<\left(\sqrt{3}\right)$

Taking intersection, we get $xϵ(1,\sqrt{3})$