Expert: Daniel Mazur - 6/7/2009

Question
Hi Daniel,

I'm asking all mathematics questions here. I know. But I really like your explanations and would like to ask you math stuff too. I hope it doesn't bother you much.

Ok, so i searched the web a bit about this number 'e' and all the sites out there give me mixed info.
Most of them have info. on just the history of the number and not so much on the number itself.

All i've sort of understood so far is that this number has got something to do with natural growth processes, that things grow in a fixed pattern or a propotion probably that is related to this number.
Is that what it is all about??
Also why do we take the logarithm to the base e for many quantities???

Would be great if you could give me how it got into place, historically, not much details,just the key points.

Also i read on some site while i was surfing, that logarithm wasn't understood earlier the way we understand it today.
I would like some info. about this too.

Thanks,
Shikhin

Answer
Dear Shikhin,
forgive me the delay, please, I have had a lot on my plate lately. As far as I remember, the number e has been found as the base of a unique case of an exponential (and logarithmic, the inverse of exponential) function. It starts with the "derivative" operation of a general exponential function f(x) = a^x. The derivative of this is f'(x) = ln(a)*a^x. You see, while a^x doesn't have anything to do (seemingly) with number 'e', in the derivative 'e' comes up as the base of the logarithm ln(a). The connection arises with realization that a general exponential a^x can be expressed via natural exponential and logarithm:

Let's define a=e^b (for any 'a' we can find a unique 'b') and logarithm the equation with base 'e' (base of exponential on the right). Then we get ln(a)=b. The general exponential then can be written f(x)=a^x=(e^b)^x=e^(b*x)=exp(bx)=exp(x*ln(a)). As the 'a' is a constant (as opposed to variable x), we have shown that all exponentials are, somehow, one! At the time, when nobody knew about 'e', people already knew they can transform exponentials and logarithms from one base to another and were looking for some kind of "default" base, in which all calculus could be simplified. Initially by numerical approximations, they arrived at the number 'e' as the base of such exponential that its derivative equals the original function. That is, how we today learn about the significance of 'e': [e^x]'==e^x. It is self-consistent, because for general f(x)=a^x the derivative [a^x]' = ln(a)*a^x and if a=e then ln(a)=ln(e)=1. This is the uniqueness of e^x and therefore 'e' itself.

The exact definition of e arose from the approximate numerical approach as the sum of terms {x^n/n!} where n goes from 0 to infinity. I found that Wikipedia here http://en.wikipedia.org/wiki/Exponential_function gives the same explanation and it's even better that from me, because of the available mathematical symbols. No-one knows, why the result is the value of 'e' we know and not another - it is simply a property of the maths that describes our Universe.

The reason for us to find exponentials in many natural laws is, that the magnitude of many effects is proportional to the magnitude of the cause. Let's say you try to describe the number of rabbits breeding in time over several generations - we will forget now, that the number N is not real, it is discreet (integer), but if we already have many rabbits to start with, this is negligible. Let's call the total number N, the increase of this number DN and the time period over which we determine the increase DN be Dt. Let the rabbits have a birth rate C (number per unit time), then the absolute rate at which they breed is DN/Dt = C*N. In other words the change is proportional to the absolute number N. This is an ordinary differential equation and can be solved by direct integration:
DN/N = C*Dt -> ln(N1)-ln(N0) = C*(t1-t0) -> N1=exp(ln(N0)+C*(t1-t0))=N0*exp(C*(t1-t0)). If we put t0=0, then N0=N(t0)=N(0), and we can change the symbols to the usual form N1->N, t1->t:
N(t) = N0*exp(C*t).  This is, why we call its logarithm "natural", I think: it is the e^x exponential that arises from our integration process of such a natural phenomenon as breeding of organisms.

To the point of natural logarithm meaning now something different than before... It doesn't. I think you have read about "Napierian logarithm", which is defined as something different than natural logarithm. I think the Napierian logarithm is the historical remnant of those approximate attempts of people at finding 'e'. As the base of NapLog is close to 16*e, I think it had been used before the value of 'e' was ascertained to a good-enough precision. And because people are people, many started to use the then-traditional term "Napierian logarithm" as a synonym of "natural logarithm". Mathematically these two are different and always will be.

I hope this helps, enjoy!
Daniel