Solve: $x^x = x\quad (*)$
I can only solve it when $x > 0$
$(*)\Leftrightarrow \ln(x^x) = \ln x$
$\Leftrightarrow x\ln x - \ln x = 0$
$\Leftrightarrow \ln x(x -1) = 0$
Then $x = 1$
How can I solve it with $x \leq -1$ ?
If it's possible, please show me how to find its domain also.
Thank you for helping!!!