Generating Intuitions for Exponential Growth

Applied Divinity Studies:

You’ve probably heard of the Rule of 70:

To estimate the doubling time of an exponential function, just divide 70 by the growth rate.

For some rates, this works really well. At 2% annual growth, the rule gives 35 years, and the actual value is 35.003 years. Other times it fails horribly. At 70% growth, the rule predicts doubling in one time step, but it actually takes 1.3.

How does the heuristic perform in general? Not that well. It’s accurate at 2% growth, but then quickly converges to being off by 0.3 timesteps.