Physicist: “e” shows up on its own a lot, and the frequent appearance of the natural log, “ “, follows from that. However, when you start using derivatives and integrals (calculus) you find that e and the natural log are indispensable and surprisingly natural. The other stunningly important property (actually tied up with the calculus property), is that e shows up in Euler’s equation, . But this means that the anti-derivative of 1/x is ln(x) (which is good to know) and not some other less natural logarithm. So, anytime you want to find the integral of 1 over some polynomial you’re going to see lots of natural logs.