In a previous post, I derided Ray Kurzweil for showing a straight line on a log-log graph and calling that an "exponential trend." As I said in that post:

...the math savvy should recognize that a straight line on a log-log graph is not indicative of exponential growth but instead illustrates a power law relationship. So does that chart contradict the exponential developments that are the cornerstone of his arguments?

Well, as it turns out, the chart does not contradict the exponential trend. You might ask, how can you get an exponential trend to look like a straight line on a log-log graph? The answer is very subtle, actually. Let's take a simple model of exponential growth:

Where n is something like number of inventions, and t is time (the other two parameters, n₀ and k, are there just for the sake of generalization). The key to making this a straight line in log-log space is to plot the number of inventions vs. the time until next invention. With this subtle change, the plot becomes linear on a log-log plot. The time until next invention, τ(t), is fairly simple to obtain:

When you plot τ(t) vs. n(t), you get a straight line. So I was wrong, and Kurzweil was right. I still maintain that the way Kurzweil plots this is very deceptive. The reason that he didn't plot it on a normal semilog plot (like all the others he normally shows), is that the scatter would look huge. By compressing both axes into log scale, it makes it look as if the correlation is better than it really is; as far as I'm concerned, this is a rather big no-no for serious academics. I think many peer-reviewers would recoil upon seeing data plotted in such a way.

Just my two cents.