How to solve $kx=e^x$

Question

How do I solve equations of the form $kx=e^x$ for $x$?

E.g. how would I solve $3x=e^x$?

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I have tried using logs

$$ \begin{align*} 3x&=e^x \\ \ln3x&=x\\ \ln3+\ln{x}&=x\\ x-\ln{x}&=\ln3 \end{align*} $$

but it doesn't look like this is going to get me anywhere.

Answer 1

This kind of equation has no solution in terms of elementary functions. You have to use the Lambert W function, which is effectively defined to be the solution of this sort of problem.