Natural logarithm and e

When I was about 14, I try to learn every key on my scientific calculator. One of the keys I couldn't figure out was the [ln] / [e^x] key. I can figure out the 10^x and the logarithm key! It is 10 raised to something, and the [log] key undoes [10^x] key (They are inverse function of each other). At first I even mistakenly thought the 'l' in ln is a capital I, and call it "in x". Then I found out it stands for "natural logarithm". Ok, why is it not called "nl" instead? My guess is that it has French origin where the adjective goes after the noun it modifies. I hate it when people use "lg" for natural logarithm, especially notorious in computer science texts. But what is so natural about the the natural logarithm?

So I asked my teacher, "what is the [ln] key for?"
"It is the natural logarithm, which is the logarithm of base e"
"What is this e?"
"See that [e^x] key? Evaluate e¹. That gives you the value of e"
"It is 2.718281828. But why are we interested in the logarithm of e? What is this number?"
"It is the sum of the reciprocals of all the factorials. e = 1+1/1! + 1/2! + 1/3! + ..."
"Why use this as a base of logarithm?"
"This base comes naturally, you will learn about it in calculus"

I punched a few terms 1+1/1! + 1/2! + 1/3! + 1/4!+... on the calculator and it is indeed getting close to 2.718281828...

Ok, some years passed, I survived calculus (and many years passed). The natural log: it turns out to be the integral of f(x) = 1/x. When the power rule fails in integrating a polynomial, the natural log come in. (See any calculus text for details)

e^x is amazing in a lot of ways. It is the only function whose derivative is itself, because e is the constant defined so that is so. It just turns out to have that sum of reciprocal of factorials representation.
Try this on your calculator, let h be a small positive number like 0.000001. Evaluate: e = (1+h)^(1/h). Yes I am trying to determine a limit as h approaches to 0 here.

Ok, it is a bit lengthly to discuss full details for that in a blog article. The main idea is that we want to evaluate the derivative of a constant a raised to the x power using the derivative's definition, you know: lim h->0 (f(x+h)-f(x))/h, and f(x)=a^x... It turns out to be a constant multiple of a^x... That limit above determines this a so that the derivative of a^x=a^x(See a good calculus book for more details)

The neat sum of reciprocal of all factorials representation of e comes from Taylor series for e^xand the "go ahead, I am not afraid of derivatives" property of e^x.

What is it good for? Scientists love to use it for modeling population growth. It also appears in compound interest and probability. Wiki has a ton of info on e.

OMG, the number e is also related to π and the imaginary number i as shown in the number one mathematicans favorite Euler's Formula: e^iπ + 1 = 0. The letter e is chosen for the big mathematician Euler.

About scientific calculator keys, another set of mysterious keys are the hyperbolic trig functions. I never find a good use for these buttons, but even if your favorite soft drinks spilled and destroyed those hyperbolic keys you can still express them with the [e^x] key. See here for details. Ok, it takes some differential equations to knowledge see how it comes in. See here if you are interested. I am not that interested in hyperbolic trig functions.

Share the excitement of this remarkable number e?