Natural logarithm: Why e?

[Loren Booda]

Their must be over a million definitions involving the constant "e". What I would like is a description of the natural logarithm in natural terms, not just saying "e is where

e
[inte] dx/x=1, etc."
1

In other words, why choose this function to define e, and how does it most fundamentally relate to other uses of e?

 

[Brad_Ad23]

That's a good question. I'm sure I could answer it where it not so late, I'll think into it.

 

[instanton]

e shows up everywhere is nature. Essentially it describe any phenomena that includes continuous build-up with certain rate. i.e. interests of your bank account, spiral pattern of shell of certain sea creatures. You can easily see from the definition of exponential function.

e = lim (1 + x) ^(1/x)

where limit takes x to zero.

Instanton

 

[Brad_Ad23]

Yes, but that doesn't really explain the question.


One thing that I remember is that for the graph of e^x, the function is its own slope at any point if that helps some insight.

 

[instanton]

Then, I don't know what the question is. e is just a trancedental number. Significance, or usefulness comes from the fact that it shows up everywhere in nature. Case for a pi is similar.

Instanton

 

[newton1]

actually the first function we define is not e
it's natural log
integrate from 1 to x for dt/t = ln(x)-ln(1)= ln(x)
because after we define natural logarithm funtion
we fine the base for the ln function
so we defined the exponatial e
hope this can help

 

[damgo]

The general idea is we very often have cases where

rate of change of X proportional to X

eg X'(t) = c*X(t). For example this very often occurs when X is a number of objects which are independent of one another: reproducing bacteria, decaying atoms. The most mathematically natural case is where c=1; though physically, there is nothing special about that unless the units of X and t are comparable.

It turns out the solution to this differential equation has the form of an exponential, X(t)=e^t, with e itself.

 

[Fermat]

Summation the reciprocal of k factorial from 1 to infinitely large is also a representation of e , but it comes from the Talor's theorem and the derivative of e^x.

 

[dg]

The first definition I was given for e is as a limit for n->oo of (1+1/n)^n.

In this form it occurs also in other scenarios like the calculation of continuous interest in finance...

As far as considering it as a natural base for logarithms... you realize how natural it is only once you start calculating derivatives I guess...

Another possibility is to say that e is the basis of logarithm when
lim (log(1+x))/x=1
x->0
that is the logarithm that goes to zero as linearly as x does.