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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How do Mathematicians determine when to use the constant  e ?I&#039;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 &quot;e&quot;. So it got me wondering; how/when/where/why do they decide to use it? I&#039;m assuming they don&#039;t build a model and include &quot;e&quot; just because, and there must be some sort of fundamental intuition behind when its use is appropriate.

darkflamexivcr · Accepted Answer

Actually, on the contrary - often you do use e &quot;just because&quot;. 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&#039;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&#039;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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