# Natural logarithm: Why e?

• Loren Booda
In summary, e is a constant that shows up in various natural phenomena involving continuous growth or change. It is defined as the limit of (1+1/n)^n as n approaches infinity, and is often used as the base for logarithmic functions due to its relationship with continuously changing values. It is also the solution to the differential equation where the rate of change is proportional to the value itself. Its significance and usefulness lies in its prevalence in nature and its mathematical properties.

#### 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?

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

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

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

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

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

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

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.

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.

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.