Question

Solve the following equation:
$x^{\log x}=1000x^2$

Answer 1

$x^{logx}=1000x^2\Rightarrow logx\left(logx\right)=log\left(1000\right)+{logx}^2\Rightarrow (logx{)}^2=3+2logx\Rightarrow {\left(logx\right)}^2-2logx-3=0\Rightarrow {\left(logx\right)}^2-3logx+logx-3=0\Rightarrow (logx+1)(logx-3)=0$

$logx=-1$ or $3.$

$x=\frac{1}{10}$ or $1000.$