I have this equation $x^x=(1-x)^{(1-x)}$ and I want to find x. The solution x=1/2 is clear and from the function plots ploted with, for example, wolfram alpha [1], is obvious this is the only situation. But how can I prove that this is the only one? I tried logarithmizing the equation and ariving at $x lnx =(1-x) ln(1-x)$, but I still couldn't prove it.

Thank you in advance! 

[1] <https://www.wolframalpha.com/input/?i=x%5Ex%3D(1-x)%5E(1-x)&wal=header>