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The visual “knee” is purely a result of your chosen scale; there is no scale-independent mathematical definition.

Put it on a log/log chart, and it’s a straight line; no part of the growth is qualitatively different than any other.

I'm well aware of that -- except that an exponential is a straight line on a semi-log chart. But to people hearing the numbers, who aren't familiar with the phenomenon of exponential growth, a linear scale, despite being somewhat arbitrary, better captures the psychological impact. Consider an idealized example: we start with one case, and a doubling time of three days. It takes the whole first month to get to about 1000 cases. Even though you and I know that that's an extremely aggressive growth rate, 1000 cases still doesn't sound like a lot to many people, when they compare it to the population of the US (or annual numbers of deaths from various other causes, like swimming pool accidents -- a comparison I've actually heard made). In the next month, though, it would get to 1M cases, which is a much larger fraction of the population and is correspondingly likelier to get people's attention.

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