Logarithmic-like growth with a limit

Question

I'm creating a curve that needs to have logarithmic-like growth but has a limit of 1. Logistic growth does not make sense in the context of my problem, and something like $P(x) = 1-e^{-bx}$ or a Hill function produces too severe an "elbow". What sort of function could I use for this issue? Alternatively, what tuning parameters would soften the curves of a Hill function or exponential or similar?

Answer 1

I'm not too sure what you mean by "logarith-like growth", but for the Hill function, wikipedia would seem to show that you can adjust the parameter $n$, and then shift the function over to the point where the "elbow" occurs to make it look more like a logarithm with its concavity. Link

Answer 2

Here's a Desmos graph I created for a previous question. You can play with the values of $a$ and $b$ using the sliders on the left side. You might find something you like there.