Question

solve $x^{{\log }_{3^x}}=9$

Answer 1

$x^{{\log }_{3}x}=9$

Taking $log$ as base $3$ on both sides.

${{\log }_{3}x}^{{\log }_{3}x}={\log }_{3}9$

${\log }_{3}x{\log }_{3}x={\log }_3{\left(3\right)}^2$

${\left({\log }_{3}x\right)}^2=2$

${\log }_{3}x=\sqrt{2}$

Taking antilog on both sides (Base $3$)

$x=3^{\sqrt{2}}$