"How do Mathematicians determine when to use the constant e ? I'm only an undergrad so sorry if this is a dumb question, but I was studying Poisson distribution and it struck me that so many models involve ""e"". So it got me wondering; how/when/where/why do they decide to use it? I'm assuming they don't build a model and include ""e"" just because, and there must be some sort of fundamental intuition behind when its use is appropriate."

blogswput 2022-09-14 Answered
How do Mathematicians determine when to use the constant e ?
I'm only an undergrad so sorry if this is a dumb question, but I was studying Poisson distribution and it struck me that so many models involve "e". So it got me wondering; how/when/where/why do they decide to use it? I'm assuming they don't build a model and include "e" just because, and there must be some sort of fundamental intuition behind when its use is appropriate.
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Answers (1)

darkflamexivcr
Answered 2022-09-15 Author has 14 answers
Actually, on the contrary - often you do use e "just because". Or, well, almost. The thing is, any exponential expression can be written as a power of e (for example 2 3 x 2 is just 1 4 e 3 x ln 2 ). And powers of e are super convenient for all sorts of reasons - the first being that d d x e x = e x and e x d x = e x + c, so powers of e are ridiculously easy to differentiate and integrate. What that means is that whenever a mathematician is building a model that involves some sort of exponential, they'll usually convert it into a power of e unless they have a pressing reason not to.
EDIT: To walk through that example:
First, remember that a b + c = a b a c , and that for any a we have a = e ln a (this isn't a fancy property of e, just logarithms in general).
So
2 3 x 2 = 2 3 x 2 2 = ( e ln 2 ) 3 x 2 2
Next, use the fact that ( a b ) c = a b c , and that 2 2 = 1 4
( e ln 2 ) 3 x 2 2 = 1 4 e ( ln 2 ) 3 x
Finally, rewriting it to be a little more readable, we get 1 4 e 3 x ln 2
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