I am trying to create a model of an economy for a video game. The game will have a population of finitely many players. I want to choose an exponential wealth distribution function that outputs the wealth of the $x$-th player
$$w(x) = c\cdot e^{rx}$$
subject to constraints on the wealth of the poorest player, the wealth of the richest player, and the total wealth of all players:
$$\begin{cases} c \cdot e^r = w_{min} \\ \\ c \cdot e^{rP} = w_{max} \\ \\ c \cdot e^{r}\bigg ( \frac{1-r^P}{1-r} \bigg ) = W \end{cases} $$
All the variables are positive real numbers. I would choose values for $w_{min},\ w_{max},$ and $W$, and I need to solve for $c,\ r,$ and $P$. Is there a closed-form solution for this? I can get close using approximations; e.g. if $r$ is small, then $c \approx w_{min}$, but I would prefer a closed form exact solution if possible.
My attempt:
Solve (1) for $r$ to get $r = \ln \big ( \frac {w_{min}}{c} \big ) $. Then divide (2) by (1) to obtain $P = \frac {\ln \big(\frac{w_{max}}{w_{min}}\big ) + r}{r}$. Then we can plug the expression for $r$ into the most recent equation to give $P = \frac {\ln (w_{max} ) - \ln (c) }{\ln(w_{min}) - \ln (c) }$. But using log properties, that simplifies to just $P = \ln \big ( \frac {w_{max}}{w_{min}} \big ) $. Then I get stuck trying to find the value of $c$. If I just plug in what I already have to (3) and try to simplify, I get
$$ \ln \bigg ( \frac {w_{min}}{c} \bigg ) ^ { \ln \big ( \frac {w_{max}}{w_{min}} \big ) } + \frac {W}{w_{min}} \cdot \ln \bigg ( \frac {w_{min}}{c} \bigg ) + W - w_{min} = 0 $$
which is a mess I can't solve.