Question

Solve the following equation:
$\sqrt{{\log }_9\left(9x^8\right)\cdot {\log }_3\left(3x\right)}={\log }_3{x}^3$

Answer 1

$\sqrt{{\log }_9\left({9x}^8\right).{\log }_3\left(3x\right)}={\log }_3{x}^3\Rightarrow \sqrt{(1+{\log }_9{x}^8)(1+{\log }_{3}x)}={3\log }_{3}x\Rightarrow \sqrt{(1+{4\log }_{3}x)(1+{\log }_{3}x)}={3\log }_{3}x\Rightarrow 1+5+{\log }_{3}x+4{{(\log }_3^x)}^2=9({{\log }_{3}x)}^2\Rightarrow 5({{\log }_{3}x)}^2-{5\log }_{3}x-1=0\Rightarrow {\log }_{3}x=\frac{5\pm \sqrt{45}}{10}\Rightarrow x=3^{\frac{5+\sqrt{45}}{10}}$