How do you solve $x^x=x$

Question

I have tried to solve this by taking the natural log of both sides and got $x\ln(x)=\ln(x)$ and after I subtracted $\ln(x)$ from both sides I got $x\ln(x)-\ln(x)=0$ which using the distributive property becomes $(x-1)\ln(x)=0$ so either $x-1=0$ or $\ln(x)=0$ and in both cases $x=1.$ what I am confused about is that $(-1)^{-1}=(-1)$ and when I solved the equation I did not get this answer. Can somebody please tell me where I messed up and how to properly solve this equation?

Answer 1

When you take the natural logarithm of both sides you need to put the operand in an absolute value. Then your final equation would be $\ln|x|=0$ $\implies$ $x=1$ or $x=-1$.