How does logistic growth rate coincide with the slope of the line in the exponential phase of the growth? For a logistic function $$f(x) = \frac{L}{1 + e^{-k(x - x_0)}},$$ people call $k$ the logistic growth rate. Now, I have encountered this statement: In the log scale the logistic growth rate coincides with the slope of the line in the exponential phase of the growth. If this is correct, how can one prove it?

Here are some exponential curves:

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*$e^x$ in cyan

*$e^{2x}$ in pink

*$1-e^{-x}$ in green

*$1-e^{-2x}$ in orange


In a sense on the left the pink curve has twice the exponential growth rate of the cyan curve, and symmetrically on the right the orange curve has twice the exponential growth rate of the green curve though in a negative sense.  When $x=0$ this translates into one slope being twice the other ($1$ and $2$ at that point), though not at other points. If you translated the curves horizontally, for example looking at $e^{x-5}$ and $e^{2(x-3)}$, then they would cross at a different point in the curves, here $x=1$, and at that point one slope would be twice the other ($e^{-4}$ and $2e^{-4}$ at that point).
Now add two logistic sigmoid curves

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*$\dfrac{1}{1+e^{-x}}$ in blue

*$\dfrac{1}{1+e^{-2x}}$ in red


Note the blue curve is close to the cyan curve and the red curve is close to the pink curve for large negative $x$, while the blue curve is close to the green curve and the red curve is close to the orange curve for large positive $x$, so in those parts of the chart sharing their growth rates.  Again when $x=0$ this translates into one slope being twice the other $\big(\frac14$ and $\frac12$ at that point$\big)$, though not at other points. Again, if you translated the curves horizontally, for example looking at $\frac{1}{1+e^{-(x-5)}}$ and $\frac{1}{1+e^{-2(x-3)}}$, then these would cross when $x=1$, and at that point one slope would be twice the other $\Big(\frac{e^{-4}}{(1+e^{-4})^2}$ and $\frac{2e^{-4}}{(1+e^{-4})^2}$ at that point$\Big)$.
So it could be reasonable to suggest the red curve in some sense has twice the logistic growth rate of the blue curve.
