Solve $x - \ln(x) > 1$ without using derivatives I would like to solve $x - \ln(x) > 1$ without studying the function $f(x) = x - \ln(x) - 1$ and computing its derivative.
I wonder if solving for $x$ in $x - \ln(x) = 1$ is possible.
It could be possible to consider the Lambert $W$-function, since we can rewrite the expression such that we have
\begin{align*}
  \ln(x) - x   &= 1  \\
  \ln(xe^{-x}) &= 1  \\
  xe^{-x}      &= e  \\
  -xe^{-x}     &= -e
\end{align*}
Yet it seems impossible and non-elementary to have a "nice" analytical solution to this.

I think you made a mistake in your solution
\begin{align*}
  x-\ln(x)  &>1  \\
  \ln\left(\frac{e^x}{x}\right) &> 1  \\
  \frac{e^x}{x}    &= e  \\
  e^x    &= xe
\end{align*}
So, it has only one solution which is $x=1.$ I hope that helps
