Thread: e

Hello community! I am mastermind. Hope I will find some pleasure here and can discuss many things.
Here my question now:

Why exactly is e called the "natural" logarithm?

And why is e^z = 1 + z/1 + (z^2)/2 + (z^3)/3 + ... ?

Thank you for help!

Hi mastermind

1. e is not the natural logarithm, it is the natural base for exponents in calculus though. e^x = 1 + x + x²/2! + x³/3! + .... and not what you posted btw. You forgot the factorials in the denominator
   
2. Log(1+x) = x - x²/2 + x³/3 + .... is the expansion for the natural logarithm that you asked for.
   
3. Both of these are examples of taylor expansions, do you know what those are or should i go through the steps?
   
4. The "natural" statement comes from the fact that if f(x) = e^x then f'(x) = e^x and this is the only base for which this statement holds. So it is natural to use this base (as it is natural to use radians for measuring angles) as you dont have messy constants popping out of your equations.

My own belief has always been that it is called 'natural' because it is used throughout formula dealing with growth and decay, that is 'e' can be used to predict the growth of humans, tree's, the natural cooling of water.. etc.

I thought that e's were drugs ?

Originally Posted by leohopkins

I thought that e's were drugs ?

From the quality of your post's I'd say you have a considerable knowledge of the subject! 8)

Hey, it wasnt ME that said that quarks could come in "flavours"!.