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Adding a cap to a logarithmic regression

I'm running a log regression to model out reach curves. A typical reach curve looks like: reach curve

So using a log regression seems the obvious choice. The reach curves are based on historical activity, but I have some additional knowledge of the total user count, so I know for certain that the reach can never exceed a certain number.

I am having a hard time blending a modelled log curve with a known hard cap. Ideally I think I should add some modifier that increases the diminishing effect as the output approaches the known cap, to help preserve the curve on the lower values (where most the activity happens) while adding in the known cap to prevent the model from returning values where the reach is 125% of all users.

Currently i'm using scipy optimize, i've added a ton of synthetic data at the known max to try and force this model to respect the max, but I think i'm going down the wrong road here:

current model

I feel like I need another function that takes the output of this one, and adds a penalty increasing in strength as the value approaches the known max, but I can't seem to get it to work, my curve keeps going negative or the blend comes out really jarring like a hard cap, any thoughts on how to approach this?



source https://stackoverflow.com/questions/77448229/adding-a-cap-to-a-logarithmic-regression
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