TITLE: Solving the equation $x e^x = e$
QUESTION [7 upvotes]: I know that $x e^x = e$ means $x = 1$, but how do you solve for it?

REPLY [7 votes]: With logarithms.
Firstly  $x>0$, Then :
$xe^{x}=e \Rightarrow \ln (xe^{x})=\ln e \Rightarrow \ln x + x = 1 $.
So:
If $x>1$ then $ \ln x >0$, thus $\ln x +x > 1+0 =1$  (contradiction)
If $x<1$ then $ \ln x < 0$, thus $\ln x +x < 1+0 =1$  (contradiction)
Thus $x=1$