How to detect exponential growth

The short answer is by drawing a straight line on a log plot.

However data is fuzzy and are you sure you didn’t just draw a tangent? To answer this question it is helpful to consider the properties of exponential growth. Exponential growth requires infinite resource as it grows to infinity. In a finite world this means it grows until it comes to an abrupt stop as all the available resource is used up. If something were to grow exponentially then not there would have to be an event happening to change the track of the curve. This may be possible under laboratory conditions but it is hard to imagine it happening otherwise. Some examples of non-exponential growth follow.

Epidemics that result in herd immunity.

Nuclear explosions resulting in unused fissionable material in the fallout.

In finance Geometric Brownian Motion is a model in which stock prices inflate on average exponentially. That this model is wrong is due to observation rather than a Mathematical argument. GBM is the basis of many financial models including the Black-Scholes model for pricing options that won the Merton and Scholes the 1997 Nobel Prize in Economic Sciences. Long Term Capital Management employed the future Nobel Laureates and had such confidence in the model that the US Federal Reserve had to intervene in 1998 after billions in losses. Subsequently traders talk of fat tails and black swans as to why the base model is wrong.

Finally an example of actual exponential growth. The compound interest paid by your bank on your savings.