Intuition behind natural log and e
The following are a set of nice explainers about intuition for what $e$ and $\ln$ mean, and why we use them constantly to discuss growth rates. You can surf through them to refresh your use of them, but I’ll highlight something the sites bring up a lot.
$e$ tells you how much growth has occurred after a given amount of time. So $e^{gt}$ tells you by how much something is scaled up after growing at a rate $g$ for $t$ periods.
$\ln$ tells you what it takes to reach a given amount of growth. So $\ln(x)$ will tell you the combination of growth rate $g$ and time $t$ that it takes to scale something up by a factor of $x$.
I think this post is the best one overall: How To Think With Exponents And Logarithms
The rest of these are also useful:
An Intuitive Guide To Exponential Functions & e
Demystifying the Natural Logarithm (ln)