What is the value of $x$ if $x^x = x$?

Question

What is the value of $x$ if $$x^x = x?$$ Can somebody show step by step please. Thanks!

Answer 1

Given:$\;\;$$x^{x} = x\;,\;$

Taking the logarithms on both the sides of equation we get

$ x\times\log (|x|) = \log (|x|)$

$ \therefore \;\: (x - 1)\times\log (|x|) = 0$

For the above equation to be true

Either $\;\;$$x-1 = 0\;\;$ or $ \;\;$$\log (|x|) =0$

Therefore $\;\;$$x = 1\;\;$ or $\;\;$$|x| = 1$.

Hence, the solution is $\;\;$$x = 1\;\;$ or $\;\;$$x = -1$.

Answer 2

Clearly, $x\ne0$

So, we have $$x^{x-1}=1$$

For $a^b=1$

either $a=1$

or $a=-1,b$ even

or $b=0$