This is more a general big picture conceptual question. I observe in life some phenomena display exponential growth. Wealth accumulation or growth of a new business or success as an artist can be included in this bucket. For years, visible signs of success may be slow, and then boom, things exponentiate.
I know some growth is logarithmic. You just can't get past a plateau. There's a kind of natural limit to a level of success. It strikes me that all things that grow exponentially must hit a limit or ceiling. You could also take the case of a foreign species introduced in a new environment in which they thrive. Their growth will be exponential, until it hits the upper limit of the carrying capacity.
Is there a type of function that combines these two concepts? It strikes me that indefinite, infinite exponential functions don't or can't exist in nature, except in rare cases (expansion of universe maybe).
I was wondering if such a thing exists and what the name for such a hybrid function would be, and where I can read up on it more, and how might it be described symbolically. I have never thought of merging or joining two separate types of functions and now I'm really curious.
Of interest can be digamma function $\psi(x)$.
On the left side it resembles $-\pi\cot \pi x$ while on the right side it resembles logarithm.