Find the constant!!!

$x$ is nothing but the Omega Constant .....

$\log_e(\frac{1}{x})=x \implies e^{\log_e(\frac{1}{x})}=e^x \implies \frac{1}{x}=e^x \implies xe^x=1$

By definition of the Lambert-W function : If $xe^x=n$ then $x=W(n)$

In this case $xe^x=1$ so $x=W(1)$ also known as Omega constant

Here is the approximation of the Omega constant up to fourth decimal digit: $\boxed{\large{0.5671}}$