Need help with $e^x=1/x$

Question

I've tried everything. I expressed $x$ and I got $x=\ln{1\over x}$, and don't know what to do. Original question is to find $e^x-{1\over x}=0$. There is a solution I've typed it in Wolfram

Answer 1

There is no closed solution. However note that:

$$e^x = \frac{1}{x}$$

has one real root. To see this you can consider it as a function , differentiate , determine the range etc. Therefore we have a root.

This root is a famous constant denoted as $\Omega$. Its approximate value is $0.5671$.

Otherwise, this root can be expressed via Lambert W.That is if $r$ denotes the root then we have $r=W(1)$.

P.S There exists an integral represantation of $\Omega$.

Answer 2

This can be rewritten as

$$ xe^x = 1$$

And here, there is no "simple" answer, you need to introduce the Lambert W function.