Intersection of functions $\ln(x)$ and $\frac{1}{x}$

Question

How to find $x$ such that $$\ln(x)=\frac{1}{x}$$

Thank you!

Answer 1

$$e ^{\ln x } = e^{\frac{1}{x}} \Rightarrow \frac{1}{x} e^{\frac{1}{x}} = 1\Rightarrow x = \frac{1}{W(1)} $$

Where $W(1)$ is the Omega constant.

Answer 2

$$\begin{align*} \ln x &= \frac1x\\ x &= e^{1/x}\\ 1 &= \frac1x e^{1/x}\\ W(1) &= \frac1x\\ x &= \frac1{W(1)} \end{align*}$$ Where $W$ is the inverse function of $we^w$.

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Alternatively, $$\begin{align*} x\ln x &= 1\\ (\ln x)e^{\ln x} &= 1\\ \ln x &= W(1)\\ x &= e^{W(1)} \end{align*}$$ and WolframAlpha confirms $e^{W(1)} = \frac1{W(1)}$, which of course satisfies $\ln x = \frac1x$.