Equation with logarithms and absolute values

Question

I have this equation and I want to solve it for $x<0$. $$\frac{\ln|x|}{|x|}=\frac{\ln|x|}{x}$$ According to WolframAlpha, the solution is $x=-1$ but I don't know how to get that.

My approach: $$\frac{\ln|x|}{|x|}=\frac{\ln|x|}{x}$$ $$\frac{1}{|x|}=\frac{1}{x}$$ $$|x|=x$$ $$-x=x$$ $$2x=0$$ $$x=0$$ which is apparently wrong. Where have I made the mistake?

Answer 1

Hint

if $x\neq -1$,

$$\frac{\ln|x|}{x}=\frac{\ln|x|}{|x|}\implies \frac{1}{x}=\frac{1}{|x|}$$