Wednesday, 21 December 2011

Musings on Euler's number and natural logarithms

In other words: a proof that

where e=2.718281828... (Euler's number)

Mainly a note to self, just so I don't have to remember it again...

Prompted by a couple of posts by Adrian McMenamin on his blog

Not entirely sure this is entirely robust or complete – but I think it’s basically the way I learned it and have remembered and rehearsed it in the past. Any comments/corrections from those more in the know welcome.

Define the function f(x) as follows:

Then f(x) is uniquely defined (for all real non-zero positive values of x) and


Also, for a variable y (independent of x but on the same domain)


where C is some constant value.

To find the value of C let x = 1. Then, since f(1) = 0

Now define the function g(x) to be the inverse of the function f(x). i.e.



And from the previous result that

we get

i.e. the function g satisfies the basic exponential identity

So, g(x) can be expressed as some number (go on, let's call it e) raised to the power of x.

Since f(x) is the inverse of g(x), f(x) can be expressed as the logarithm to the base e of x. i.e.


That's the first bit done (I think)...

...but what's the value of e?

First, look at the properties of the derivative of the function*


Remember (from above)


Remember too that

[This result allows for the derivation of both the sum of a series and limit definitions of Euler's number. What follows here is the derivation of the sum of a series definition - the limit definition is derived below]

It follows directly from the result

that all higher derivatives of

can be expressed as follows:

for any positive integer n.


is infinitely differentiable and


represents the value of the nth derivative of

with respect to x at x = 0.


can be expressed as a Maclaurin series:

So set x = 1 to calculate e to arbitrary precision using


Consider the limit




for any y and

is defined, therefore L'Hopital's rule applies. i.e.



and as



therefore (since the exponential function is continuous*)

Or alternatively if


I've used g(x) to denote the exponential function here to try and avoid preconceptions but it's obviously more commonly denoted by

I think I'm right in saying that, without a definition of the exponential function:

and exponential identities such as

can only be defined where x (and y) is rational.

However, because

is (by definition) continuous and differentiable (with non-zero derivative) at all points on its domain of all non-zero positive numbers (range is all real numbers), its inverse

is also continuous and differentiable at all points on its domain of all real numbers (its range is all non-zero positive numbers). Therefore e can be raised to the power of any real number, not just rationals. This leads on to the definition of values for non-zero positive real numbers raised to the power of any real number - and eventually can be completely extended to include negative and complex numbers. A few steps beyond that and ultimately comes the famous and beautiful (and immensely useful) Euler function and identity:

I think... I might post a proof of that too at some point. Weather, time and memory permitting.