An exponential function’s output changes at a rate proportional to its current value, which serves to model unlimited growth. I.e. the exponential function \(P(t) = \mathrm{e}^{kt}\) is the only function that satisfies the differential equation \(\frac{\mathrm{d}P}{\mathrm{d}t} = k P.\) The number \(k\) is the exponential rate; for \(k \gt 0\) its called the growth rate and for \(k \lt 0\) its called the decay rate. Note that \(k\) will be an instantaneous rate-of-change, and will be different from the rate-of-change over some duration.
A logistic function’s output changes at a rate proportional to the product of its size and TK.... I.e. the logistic function \(P(t) = \frac{L}{1+C\mathrm{e}^{-Lkt}}\) is the only function that satisfies the differential equation \(\frac{\mathrm{d}P}{\mathrm{d}t} = k\bigl(P\bigr)\bigl(L-P\bigr)\) for \(L \gt 0\) and \(k \gt 0.\)
Limited / restrained growth \(\frac{\mathrm{d}P}{\mathrm{d}t} = k\bigl(L-P\bigr)\) \(P(t) = L + C\mathrm{e}^{-kt}\) for \(L \gt 0\) and \(k \gt 0.\)