TITLE: How to solve $\ln x+x=1$
QUESTION [7 upvotes]: How can I solve this equation: 
$$\ln x+x=1$$ 
We had it on a local Olympiad math contest problem.

REPLY [5 votes]: Intuitively
\begin{align}
\ln x+x&=1\\
\ln x+x\ln e&=1\\
\ln x+\ln e^x&=1\\
\ln (xe^x)&=1\\
xe^x&=e^1\\
xe^x&=1\cdot e^1
\end{align}
By comparing LHS and RHS, we will obtain $x=1$.