Question

The total number of integral value of $x$ satisfying the equation $ x^{log _{3^{x^2 +\left(log_3 x\right)^{2 -10}}}} =1/x^{2}$ is

Answer 1

Taking log of both the sides with base $3$,
$\Rightarrow \left(log_{3} x^{2} +\left(log_{3} x\right)^{2} -10\right)\left(log_{3} x\right) = -2 log_{3} x$
$\Rightarrow log_{3} x\left[log_{3} x^{2} +\left(log_{3} x\right)^{2} -8\right] =0$
$\Rightarrow log_{3}x=0$ or $2 log_{3}x+\left(log_{3}x\right)^{2} -8 =0$
$\Rightarrow x =1, log_{3} x = - 1\pm3$ i. e. $log_{3} x =2, log_{3} x= -4$
Hence $x = 1,3^{2},3^{-4} = 1, 9, 1/81$.