How do I interpret a log-plot that looks exponential?

Question

When looking at an x,y plot such that the y-axis is the logarithm of $f(x)$, generally the plot looks like a line. How am I supposed to interpret a plot that still looks exponential after applying a log on the output?

Answer 1

Your observation means that the function is growing faster than exponentially. You might consider taking logarithms again.

This image from wikipedia shows that the logarithm of the number of digits of the largest known prime $p$ is increasing linearly with time. But the number of digits of $p$ is the logarithm of $p$, so if you just graphed the logarithm of $p$ over time it would grow exponentially.