I wrote a post about distinguishing between logistic curves, specifically for finding their turning points.

That post was highly mathematical; but here is a visual "proof" of the "theorem":

• Figuring out the turning point of a logistic curve before hitting that turning point is bloody hard, mate.

"Proof": The following is a plot two curves:

1. The logistic curve $1/(1+e^{-x})$ up to its turning point at $x=0$.
2. The exponential curve $0.51e^{0.69x}$, which never has any turning points.

So, if the data was noisy, could you distinguish between the curve that's reached its turning point, and the one that will never have one?

Things get even worse if we stop before the turning point; here's the plot of the logistic curve up to $x=-\log \left(3\right)\approx -0.48$, with the $y=0.25$ being half of the value at the turning point. This is plotted against the exponential $0.64e^{0.85x}$: