Question

Solve the following equation:
${\log }_x\left(9x^2\right)\cdot {\log }_3^{2}x=4$

Answer 1

${log}_x{9x}^2.({log}_{3}x{)}^2=4\Rightarrow ({log}_{x}9+{log}_x{x}^2\left)\right({{log}_{3}x)}^2=4\Rightarrow (2{log}_{x}3+2)({log}_{3}x{)}^2=4\Rightarrow 2({log}_{3}x{)}^2+2\left({log}_{3}x\right)-4=0\Rightarrow {(log}_{3}x{)}^2+\left({log}_{3}x\right)-2=0\Rightarrow {log}_{3}x=-2,1\Rightarrow x=3^{-2},3\Rightarrow x=\frac{1}{9},3.$