How to solve an equation with the unknown variable both in the exponent and in linear summand?

Question

How to solve the following equation?

$$e^{dz} - mdz = 1$$

where $z$ is the unknown variable, the others are constants, $\exp(x)$ is taking $e$ to the power $x$ . I am interested in real solutions for m > 1.

Answer 1

$z = 0$ is always a solution. Others, for $m, d \ne 0$, are

$$ z = -d^{-1} \left(1/m + W(-e^{-1/m}/m) \right) $$

where $W$ is a branch of the Lambert W function.

EDIT: I suspect you're interested in real solutions where $m$ is real. For $m < 0$, $-\exp(-1/m)/m) > 0$ so the principal branch of Lambert W can be used, but it will give $z = 0$. For $m > 0$, $-e^{-1} \le \exp(-1/m)/m < 0$ so either the $-1$ or the $0$ branch can be used; for $0 < m < 1$ the $-1$ branch gives $z=0$ while the principal branch gives a negative solution. For $m > 1$ the principal branch gives $z=0$ while the $-1$ branch gives a positive solution.