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I am in the middle of solving this problem and I came across a sub problem that sorta has the format

$$\ln(x)=cx$$

I tried looking online on how to solve for $x$ if given $c$ but I couldn't find anything that I understood fully. What I want to know is how to express $x$ in that equation in terms and $c$ and without logarithms in the final expression.

Rodrigo de Azevedo
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alienare 4422
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    Raise both sides to the power of $e$:

    $$x = (e^c)^x$$

    Then you'll have to follow the steps outlined here with $a=e^c$, $b=1$, and $c=0$ ($c$ as in the equation in the linked question, not your $c$).

    The short version, though, is that you can't solve it without the rather complicated Lambert $W$ function.

     – PrincessEev Jun 24 '22 at 18:52
  • @EeveeTrainer thank you..... I guess I will have to look into the lambert function then – alienare 4422 Jun 24 '22 at 18:58
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    You can manipulate the equation $x = e^{cx}$ to

    $$(-cx) e^{-cx} = -c$$

    which has the form $we^w$ on the LHS and no $x$ on the RHS. Then assuming the RHS is in the range of $we^w$ and the domain of the $W$ function,

    $$-cx = W(-c)$$

     – peterwhy Jun 24 '22 at 19:18

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