Have a piece of Pi

[LemmingofGlory]

Which (e) is the basis of so many functions from compound interest to radioactive decay to population growth.

Pardon me if I'm completely off the mark, but doesn't it not *really* matter what base logarithm you use when working with those problems? Just that e is often used, seeing as how it is the 'natural' logarithm's base.

The section you quoted refers to functions which are exponential (not logarithmic) in some base. (Because f(x) = e^x is injective) For any positive real number a there is a unique real number r such that e^r = a (r = ln(a)). This means that we can convert between an exponetial function in base a and an exponential function in base e thusly:

f(t) = (a)^t <=> f(t) = (e^r)^t = e^(rt)

So to answer your question: providing a > 0, it does not matter what base exponential we use, because it's equivalent to some exponential in base e. And e's prominence in growth modeling is not just because of some silly convention, but because functions that are exponential in e are amazingly easy to deal with when doing anything involving calculus.

Exponential functions with base e also come flying out of our asses when we do anything involving differential equations. In fact, the phrase "differential equations" comes to us (slightly bastardized) from the ancient Sumerian holy book e-cronomicon. The phrase means "a diarrhea of e".

-Lemmy