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

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 1/2^{2}, and 1/4! is less than 1/2^{3}, and so on, so the
whole series adds up to less than 1 + 1
+ ˝ + 1/2^{2} + 1/2^{3}
+ 1/2^{4} + … = 3.)

Taking our definition of *e* as the infinite *n*
limit of _{}, it is clear that *e ^{x}* is the infinite

(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)/*m*^{2}
goes to 1, as does *m*(*m*-1)(*m*-2)/*m*^{3}. Thus, as we take *m* to infinity, the *m*
dependence of each term disappears, leaving

_{}

_{}

so when we differentiate *e ^{x}*, we just get

_{}

and now differentiating the
series term by term we see _{}, _{}, etc., so the function *e ^{ax}* is the solution to differential equations of
the form

Instead of differentiating term by term, we could have written

_{}

where we have used _{} in the limit _{}

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

*y* = ln *x,* if *x* = *e ^{y}*

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 *e ^{y}*
unless I take the lower limit 1, since that gives for the value

*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.