Solving $(x-1)\exp x = c$ with the Lambert W Function

Question

The equation

$$x\mathrm{e}^x = c$$

has a solution

$$x = W(c)$$

where $W$ is the Lambert W function and $c$ is constant.

Can the equation

$$ (x-1) \exp (x) = c $$

be solved using the Lambert W function?

Answer 1

Notice that

$$(x-1)e^x = (x-1)e^{x-1} \cdot e$$

Therefore,

$$(x-1)e^x =c \iff (x-1)e^{x-1} = \frac c e $$

Take the $W$ function of both sides:

$$x-1 = W \left( \frac c e \right)$$

Then just add $1$ and you're done!

$$x = 1+W \left( \frac c e \right)$$