Question

Solve for $x$

$3^{{\log }_3^{2}x}+x^{{\log }_{3}x}=162$

Answer 1

$3^{{\log }_3^{2}x}+x^{{\log }_{3}x}=162$

$\Rightarrow 3^{{\log }_{3}x{\log }_{3}x}+x^{{\log }_{3}x}=162$

$\Rightarrow x^{{\log }_{3}x}+x^{{\log }_{3}x}=162$

$\Rightarrow 2x^{{\log }_{3}x}=162$

$\Rightarrow x^{{\log }_{3}x}=81$

Taking $\log$ both sides, we get

$\Rightarrow {\log }_{3}x{\log }_{3}x={\log }_{3}81$

$\Rightarrow {\left({\log }_{3}x\right)}^2={\log }_3{3}^4=4{\log }_{3}3=4$

$\Rightarrow {\left({\log }_{3}x\right)}^2=4$

$\Rightarrow {\log }_{3}x=\pm 2$

$\Rightarrow x=3^2,3^{-2}$

$\therefore x=9,\frac{1}{9}$