Can $x$ be defined without itself when $\frac{p^{x}}{x} = c$?

Question

If the equation is

$$\frac{p^{x}}{x} = c$$

where $p$ and $c$ are constants.

then find $x$.

Answer 1

There does not exist an elementary closed form solution. However, one can express it in terms of the Lambert W function.

By definition, the Lambert W function is the inverse of: $$f(z)=ze^z \tag{1}$$

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Firstly, we use the identity $p^x\equiv e^{\ln{p}\cdot x}$. This gives: $$\frac{1}{x}\cdot e^{\ln{p}\cdot x}=c$$ Now, let's substitute $u=-\ln{p}\cdot x$. Therefore: $$-\frac{\ln{p}}{u}\cdot e^{-u}=c$$ After some algebraic manipulation, we obtain: $$ue^u=-\frac{\ln{p}}{c}$$ Now, we may use the definition given on $(1)$. Thus, we obtain: $$u=W\left(-\frac{\ln{p}}{c}\right)$$ Substituting back for $u$ gives an explicit solution for $x$.