Context: The nature of base $e$. The limit derivation of $e$ deals with continuous compounding. In the physical world, nature and real life, percentage growth is often in "real time" and not artificial like monthly interest compounding.
So, let's discuss the graph of $y=ln(x)$ The theme of this graph is inverse exponential growth. Like the Richter scale. You need a huge increase in X to get an increase in Y. f(3) is exponentially higher than f(2), etc.
Ran across this today $y=ln(ln(x))$ while playing around with some integrals. What exactly does this mean? When would a function need to take a log of an existing log function?