Nothing in life is exponential, but there are a lot of people who confuse the first half of a sigmoid curve for it.
We have not yet hit the linear part of the sigmoid, and are not close. But the good modes out there take that sigmoid shape into account.
(Also, why don't Singularity advocates know about sigmoids?)
https://news.ycombinator.com/item?id=31429475
and there is just the one free parameter k. Sometimes in practice have a shot at estimating k.
They think the curve goes so far up the y axis before it flattens off that it might as well be exponential to us monkeys stuck at the bottom of the slope.
I also see, as a common trope, people talking about “the knee of the exponential”; as they clearly don’t know how the exponential function works, I have no idea what they think.
d/dt y(t) = k y(t) (b - y(t))
Solve that and get a sigmoid, lazy S, curve. Yup, the solution, easy closed form, first calculus, is in terms of exponentials.
Could use that to model, say, the growth of immunity to Covid-19. Once it was used to project the growth of FedEx -- pleased the BoD and saved the company.
Addition: Ah, will save you a little first calculus. In TeX,
$$ y(t) = { y(0) b e^{bkt} \over y(0) \big ( e^{bkt} - 1 \big ) + b} $$