Question

Solve
${2.3}^{lo{g}_3^{2}x}+x^{lo{g}_{3}x}=162$

Answer 1

$let{\log }_{3}x=z$

$x=3^z$

$2.3^{{\left({\log }_{3}x\right)}^2}+x^{{\log }_{3}x}=162$

$2.3^{z^2}+{(3}^z{)}^2=162$

$2.3^{z^2}+3^{z^2}=162$

$3^{z^2}.3=162$

$3^{z^2}=54$

$z^2={\log }_{3}54$

$z=\pm {\log }_{3}54$

${\log }_{3}x=\pm {\log }_{3}54$

$x=54,\frac{1}{54}$