# The Natural Exponential Function

**The definition. **The natural logarithm function ln x
is an increasing function whose domain

is the open interval (0,∞) and whose range is R, all of the real numbers. Since
it’s increasing,

therefore it’s one-to-one and has an inverse, the natural exponential function,
which we denote

exp x. Soon we’ll also denote it e^{x}, but as that notation already has
a meaning when x is a

rational number, we’ll have to show exp x agrees with e^{x} for rational
x.

Now, exp is inverse to ln, and that means

exp x = y iff x = ln y.

Let’s start by showing exp(m/n) = e^{m/n} for rational numbers m/n so we
can use the

usual exponential notation. We know that

But since exp is inverse to ln, that statement is equivalent to

which is what we wanted to show.

Now we can use the usual exponential notation, but be aware that exp is still
used when

the exponent is complicated. For instance, a notation like
is preferred by many

over .

**Properties of the natural exponential function.** That the natural
exponential function

is inverse to the natural log now reads

e^{x} = y iff x = ln y |

From this logical equivalence, every statement about ln
can be converted to a statement

about e^{x}.

Here’s a table with properties of ln and exp.

The first few lines in the table are direct translations
of the properties of ln into properties

of exp. The later ones are almost direct, but not quite. For example, let’s see
how the identity

ln xy = ln x+ln y implies the identity . It
would help to change the variables in one

of the identities since they don’t exactly correspond. Let’s show the identity
ln xy = ln x+ln y

implies the identity .

Let s = ln x so that x = e^{s}, and let t = ln y so that y = e^{t}.
Since ln xy = ln x + ln y,

therefore s + t = ln x + ln y = ln xy. That implies
, which equals e^{s}e^{t},
and that was

what we wanted to show.

Note that the final equation in the right column of the table,
is not quite

satisfactory. We would prefer to have for
any real number y, not just for rational

y = m/n, but we haven’t yet defined what
would mean since we have only defined

exponentiation when the base is e, not when the base is an arbitrary real number
like e^{x}.

That comes soon.

**
The derivative and integral of the natural exponential function.** Recall
that, for

positive x, , that is, . Let’s use that information to determine

the derivative and integral of e

^{x}. We’ll use the inverse function theorem for derivatives to to

that.

The inverse function theorem says that if y = f(x) so that x = f

^{ -1}(y), then

that is,

Right now, we have y = f(x) = ln x, x = f^{ -1}(y)
= e^{y}, and f'(x) = 1/x. We want to find the

derivative of e^{y}, that is, . By
the inverse function theorem, that

equals

Thus, the derivative
of the exponential function e^{y} is itself e^{y}. If we switch
to x as the

independent variable, we can write this result as the exponential rule for
derivatives

It is this property of the exponential function that gives
it lots of applications in science.

Frequently, an exponential function has an exponent that is not just the
variable x, but

a function of x, for instance . To find its
derivative, use the exponential rule for

derivatives along with the chain rule:

More generally,

We can immediately use the exponential rule for
derivatives, , to integrate the

exponential function.