ΔV is a logarithmic function, with the rocket mass in the logarithm, but we consider the inverse function here, where the rocket mass is expressed in function of ΔV. And the inverse of the logarithm is exponential.
You usually have:
ΔV = Ve × Ln(mass / dry_mass)
(Ve = exhaust speed = g×Isp)
So by inverting it:
mass = dry_mass × Exp(ΔV/Ve)
I usually write mass = fuel_mass + dry_mass, so you can also write it that way:
fuel_mass = dry_mass × (Exp(ΔV/Ve) - 1)
Also, the curve given by TtTOtW has an hyperbolic behaviour: it becomes infinite when you hit a limit. It's because when you add some fuel, you also have to consider the added dry mass from the tank. A fuel tank has a "total mass to dry mass" ratio of 10, so the "mass/dry_mass" ratio of your rocket will always be lower than that.
That's why the maximum delta-V for a stage is:
ΔVmax = Ve × Ln(10)
You would have to take this into account to get the curve showed by TtTOtW, but I already reached the quota of authorized maths per post