Visualizing exponential growth is something we're currently all doing; we look at the covid-19 graphs either worldwide or, for the most part, our own countries (China and South Korea are exceptions) and we see that we're somewhere along the way up a very steep slope in terms of new cases. Even when we're not talking about the spread of a disease, seeing a graph that has exponential growth (of anything at all) can be scary.
I came across this really neat YouTube video that gives us another way to look at exponential growth. So, let's start with the typical curve you can see if you're looking at the data from Johns Hopkins University. It's plotting cumulative number of cases on the y-axis and time on the x-axis. With very few exceptions, every country is on the steep part of the exponential curve.
What this video suggests though (and this just makes sense anyway) is to use a logarithmic scale for the y-axis (the number of new cases). This makes sense, the inverse of an exponential is a logarithm and will flatten out the curve. We could do the same for the x-axis (time) but then the graph would become very misleading as each interval would represent, for example, a doubling of the previous value. So, the first interval would represent 1 day, the next 2, the next 4, the next 8, and by the time we got to the 10th interval you'd be looking at a period of a little short of 3 years. So, scratch that, don't use a logarithmic scale for time unless you really mean it.
Now, let me go back to something I hid back there; the y-axis is used for the number of new cases. Even this is, in many countries, growing exponentially so it still makes sense to use a logarithmic scale. Now, on the x-axis, they use the total number of cases (also on a logarithmic scale). Now, points on the graph represent the ratio of new cases to existing cases and, when both scales are logarithmic comes out a a fairly straight line. Well, fairly straight until you get things under control, then the curve plummets as the number of new cases drops in comparison to the total number of cases.
Why care? Well, first of all, it's reassuring that there's a way to visualize things that's helpful. Second, there are some important caveats (e.g., time is implied by progression along the curve). But, third, and most important, this is a way to visualize any kind of exponential growth which eventually reaches a limit. Even if it doesn't reach a limit, what you'd see would be the straight line curve continuing on. If growth stops, the line drops. Simple.
I'm considering using this kind of visualization for, for example, measures of defect growth in software systems.