Equation $e^{\frac{1}{x}} - x =0$

Question

Can someone solve this equations with steps

$$e^{\frac{1}{x}} - x =0$$

I dont know how to start. I tried adding logarithms but that doesn't help.

Answer 1

This equation cannot be solved using “traditional” algebraic manipulations. In this case, one would use the Lambert W function: $$W(x): x = W(x)\cdot e^{W(x)}$$ or in other words, it is the solution of the equation $x = w e^w$.

With this knowledge, we can try to substitute $y:=\frac{1}{x}$:

$$\Rightarrow 0 = e^y-\frac{1}{y} \Rightarrow \frac{1}{y} = e^y \Rightarrow 1 = y\cdot e^y\\ \Rightarrow y = W(1)\Rightarrow x = \frac{1}{W(1)} $$

See further here

EDIT: I just want to add the following: Some people tend to say this would be a trick and no real solution, because this is like saying “I don't know the solution, so I will assign it a name” — but technically, that's exactly what is done when defining the (square) root!

Once we know that the inverse of a function exists (in this case this can be verified using the Lagrange inversion theorem), we can define some function to be the solution, even if we define it just implicitly by its properties, and call this a solution with full legitimacy.

But however, that's just my two cents.