I'm not claiming that is what you were saying, I just wanted to call out this point in case other people read this that may come to this conclusion.
FWIW, the solution to f'/f = exp(-t) is f(t)=c*exp(-exp(t)), which actually looks like an epidemiological curve, but like any sigmoid it will be hard to determine the actual saturation point until after it has already passed the inflection point.