# The Number e and the Exponential Function

## Michael Fowler,UVa

Disclaimer: these notes are not mathematically rigorous.  Instead, they present quick, and, I hope, plausible, derivations of the properties of e, ex and the natural logarithm.

### The Limit

Consider the following series:  where n runs through the positive integers. What happens as n gets very large?

It’s easy to find out if you use a scientific calculator having the function x^y.  The first three terms are 2, 2.25, 2.37.  You can use your calculator to confirm that for n = 10, 100, 1000, 10,000, 100,000, 1,000,000 the values of  are (rounding off) 2.59, 2.70, 2.717, 2.718, 2.71827, 2.718280.  These calculations strongly suggest that as n goes up to infinity,  goes to a definite limit.  It can be proved mathematically that  does go to a limit, and this limiting value is called e.  The value of e is 2.7182818283… .

To try to get a bit more insight into  for large n, let us expand it using the binomial theorem. Recall that the binomial theorem gives all the terms in (1 + x)n, as follows:

To use this result to find , we obviously need to put x = 1/n, giving:

.

We are particularly interested in what happens to this series when n gets very large, because that’s when we are approaching e.  In that limit,  tends to 1, and so does .  So, for large enough n, we can ignore the n-dependence of these early terms in the series altogether!

When we do that, the series becomes just:

And, the larger we take n, the more accurately the terms in the binomial series can be simplified in this way, so as n goes to infinity this simple series represents the limiting value of. Therefore, e must be just the sum of this infinite series.

(Notice that we can see immediately from this series that e is less than 3,  because 1/3! is less than 1/22, and 1/4! is less than 1/23, and so on, so the whole series adds up to less than  1 + 1 + ½ + 1/22  + 1/23 + 1/24 + … = 3.)

### The Exponential Function ex

Taking our definition of e as the infinite n limit of , it is clear that ex is the infinite n limit of .  Let us write this another way: put y = nx, so 1/n = x/y.  Therefore, ex is the infinite y limit of .  The strategy at this point is to expand this using the binomial theorem, as above, and get a power series for ex.

(Footnote: there is one tricky technical point.  The binomial expansion is only simple if the exponent is a whole number, and for general values of x,  y = nx won’t be.  But remember we are only interested in the limit of very large n, so if x is a rational number a/b, where a and b are integers, for n any multiple of b,  y will be an integer, and pretty clearly the function  is continuous in y, so we don’t need to worry.  If x is an irrational number, we can approximate it arbitrarily well by a sequence of rational numbers to get the same result.)

So, we need to do the binomial expansion of  where y is an integer—to make this clear, let us write y = m.

Notice that this has exactly the same form as the binomial expansion of  in the paragraph above, except that now a power of x appears in each term.  Again, we are only interested in the limiting value as m goes to infinity, and in this limit m(m – 1)/m2 goes to 1, as does m(m-1)(m-2)/m3.  Thus, as we take m to infinity, the m dependence of each term disappears, leaving

### Differentiating ex

so when we differentiate ex, we just get ex back. This means ex is the solution to the equation , and also the equation , etc.  More generally, replacing x by ax in the series above gives

and now differentiating the series term by term we see ,  , etc., so the function eax  is the solution to differential equations of the form , or of the form  and so on.

Instead of differentiating term by term, we could have written

where we have used  in the limit

### The Natural Logarithm

We define the natural logarithm function ln x as the inverse of the exponential function, by which we mean

y = ln x,  if x = ey

Notice that we’ve switched x and y from the paragraph above!  Differentiating the exponential function  in this switched notation,

,  so .

That is to say,

Therefore, ln x can be written as an integral,

.

You can check that this satisfies the differential equation by taking the upper limit of the integral to be  then x, subtracting the second from the first, dividing by , and taking very small.  But why have I taken the lower limit of the integral to be 1?  In solving the differential equation in this way, I could have set the lower limit to be any constant and it would still be a solution—but it would not be the inverse function to ey unless I take the lower limit 1, since that gives for the value x = 1 that y = ln x = 0.   We need this to be true to be consistent with x = ey, since e0 = 1.

Exercise: show from the integral form of ln x, that for small x, ln(1 + x) is approximately equal to x. Check with your calculator to see how accurate this is for x = 0.1, 0.01.