Are natural logs, natural in any sense?

Question

are natural logarithms of a number natural in any sense? Why are they called natural when the base is e and common when the base is 10?

Answer 1

Part of this involves calculus reasons, namely that the derivative of $\ln(x)$ is $1/x$, a claim other logarithms cannot make; so $\ln$ is a "unique" logarithm. Moreover, it's the inverse of $e^x$, which is also special among exponential functions, again for calculus reasons (it is its own derivative).