An exponential function is an exactly straight line on a logarithmic plot - and if the rate slows, it will go from linear to horizontal.
However, an "S-shaped" adoption curve could be a sigmoid function raised to some arbitrary power. If f(x) is a sigmoid function, then f(x)^2 will look like a sigmoid with a steeper vertical, and f(x)^100 will look like a step function (going from 0 adoption to 100% adoption in one day). So, if the power is somewhere between 1 and 2, it will look super-exponential for some period on the log plot.
(e^x)^2 = e^x * e^x = e^(x+x) which is still a straight exponential (This also applies to raising to any abitrary power)
Sorry if I wasn't clear. f(x) in my example is not e^x
f(x) = 1/(1+e^(-x))
If you raise this function to higher and higher powers, it will look more and more like a step function and have a super-exponential growth phase on a log-chart (in the limit, it will look like a step function on the log chart as well as the linear chart)
OK. Gotcha. Though I'm not sure it's good to plot that directly on a log chart as the offsets are wrong for an adoption. Probably want the Gompertz function but I have no idea how that behaves on a log chart.
