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Pavitra comments on Why are certain trends so precisely exponential? - Less Wrong
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More crucially, how many trends almost, but not quite look precisely exponential? Are precisely-exponential trends the tip of a long tail, or an additional local mode?
The logistic curve, for example, is extremely similar to the exponential curve for small values seeing as the latter is y' = ky and the former is y' = ky(1-y). That got Malthus with his whole doomsday arguments about population growth outstripping resources (at least for the time being).