Question

Solve the following equation:
$x^{3\log x-\frac{1}{\log x}}=\sqrt[3]{310}$

Answer 1

$x^{3\log x-\frac{1}{\log x}}=\sqrt[3]{310}\Rightarrow 3\log x-\frac{1}{\log x}=\frac{1}{3}\left({\log }_{x}10\right)\Rightarrow 3\log x-\frac{1}{\log x}=\frac{1}{3}\left({\log }_{x}10\right)\Rightarrow 3\log x-\frac{1}{\log x}=\frac{1}{3\log x}\Rightarrow 3\log x=\frac{4}{3\log x}\Rightarrow {\left(3\log x\right)}^2=4\Rightarrow 3\log x=\pm 2.$

Therefore, $x={10}^{\frac{2}{3}},{10}^{\frac{-2}{3}}$