I understand that the natural logarithm was developed by Gregoire de Saint-Vincent and Alphonse Antonio de Sarasa as to represent the area under the curve of the hyperbola $\frac1x$ before the development of calculus.
The base of the natural logarithm is an irrational number e first observed by John Napier but defined by Jacob Bernoulli when studying compound interest as:
$$\sum_{i=0}^n \frac{x^n}{n!}$$
Speaking for many, as a non-mathematician I can see the importance of e for modeling continuous growth why it was studied and therefore view the natural logarithm as a way to represent how many cycles of continuous growth a process has gone through.
I would like a deeper understanding of these functions, how does an applied mathematician view these functions?
Does 1/x model any real world processes?
If we did not have a need in the past to simplify calculations with logarithm tables, would logarithms be of central importance in a primary math education?
What is the importance of the natural logarithm and what insight does it give besides being the x in $ x = ln y $ from $ y = e^x $ ?