Question

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

Answer 1

Answer: 2

Taking log of both the sides with base $3$,
$\Rightarrow \left(lo{g}_3{x}^2+{\left(lo{g}_{3}x\right)}^2-10\right)\left(lo{g}_{3}x\right)=-2lo{g}_{3}x$
$\Rightarrow lo{g}_{3}x\left[lo{g}_3{x}^2+{\left(lo{g}_{3}x\right)}^2-8\right]=0$
$\Rightarrow lo{g}_{3}x=0$ or $2lo{g}_{3}x+{\left(lo{g}_{3}x\right)}^2-8=0$
$\Rightarrow x=1,lo{g}_{3}x=-1\pm 3$ i. e. $lo{g}_{3}x=2,lo{g}_{3}x=-4$
Hence $x=1,3^2,3^{-4}=1,9,1/81$.