# Exponential and Power Functions

**Definition 0.1.1** (Power Function). A power function is a
function of the form
, where

a ∈ R.

Thus, a power function is a function where the base of the exponential varies as
an input. Very

basic examples of power functions include f(x) = x and f(x) = x^{2}. Note that f(x)
= x maps

the real numbers to the real numbers, where f(x) = x^{2} maps the real numbers to
the nonnegative

real numbers. Some power functions are only defined as maps on the real numbers
for a domain of

nonnegative real numbers, such as

For this function a negative input is not defined, because complex numbers are
required to make

sense of the square root of a negative number.

Using power functions as our most basic building blocks, we arrive at polynomial
functions.

One of the most basic ways in which we combine functions is in a linear
combination.

**Definition 0.1.2** (Linear Combination). A linear combination of the functions
is

a function

where .

**
Definition 0.1.3** (Polynomial). A polynomial is a function that can be written as
a linear combination

of power functions. Thus, polynomials take on the form

where .

Polynomials provide us with a large class of simple functions to work with. In constrast to

an arbitrary function, a polynomial is very well-behaved, and as a result has a number of useful

properties. As we delve further into the study of calculus we will use polynomials to approximate

other functions, and eventually be able to represent much more complicated functions as infinite

sums of power functions. Such representations are called power series.

In contrast to power functions, exponential functions are functions where the exponent varies

as an input.

**Definition 0.1.4**(Exponential Function). An exponential function is a function

(positive real numbers), .

Note that when we are talking about exponential functions we are only interested in exponentials

with base a > 0. We are not interested in a = 1, because it is simply a constant function. Since

this constant function behaves differently from the rest of the exponential functions we will deal

with, we simply exclude it from the list of exponential functions. All of the exponential functions

have a domain of R and a range of (positive real numbers). This means that the output of

an exponential function is always positive. In fact, exponential functions are strictly increasing,

which means for each exponential there is a corresponding inverse function (see theorem ??). These

inverse functions are called logarithms.

**Definition 0.1.5**(Logarithmic Function). A logarithmic function is a function

where is the inverse function of a

^{x}.

Since exponentials and logarithms are inverses, we have

for all a > 0, a ≠ 1. By virtue of this inverse relationship, logarithms
inherit a number of useful

properties from exponentials. For instance, by using the relationship

we can deduce

Above we simply use the property that , in order to move from step 2
to 3, and step

3 to 4.

We can also deduce a rule for the logarithm of a product, noting that the
exponential function

a^{x} has a range of the entire positive real numbers
. In other words, for any
there is

some b ∈ R so that

Similarly, for any positive y, we can write y = a^{c}, for some c. As a result,

Thus, the logarithm of a product of two numbers is the sum of the logarithms.
The full table of

properties of logarithms follows.

**Theorem 0.1.2** (Rules of Logarithms). Let
. It follows that:

Property 2 above follows if we write , where m ∈ R. Using the inverse
relationship

of exponentials and logarithms we also know x = a^{m}. Thus,

Taking the base a logarithm of each side of the above equation,

The third property is simply that a^{0} = 1 for all a. Properties 4 and 5 are
written only for

convenience - they follow immediately from the previous three properties. Can
you see why?

Given that for any value of a > 0, a ≠ 1 we have both an exponential and
corresponding

logarithmic function, we have access to a plethora of functions through
exponentials and logarithms.

However, as we noted previously, we are currently unable to evaluate
exponentials for all but a very

small set of numbers. Similarly, we have difficulty in actually finding the
values of logarithmic

functions. Thus, even though we have defined this large
class of functions, and have found that

they have a number of useful properties, we still cannot actually evaluate them
in most situations.

It turns out that the solution to this problem lies in polynomial functions.
Given the difficulty

of evaluating exponential functions, we can instead turn to approximating their
values. Using the

power of calculus (through the limit) we will actually be able to represent
exponential functions

as a sum of an infinite number of power functions (with natural-number
exponents). Since it is

relatively easy to evaluate power functions, this will give us a means of
accessing these much more

elusive exponential functions. We will deal with logarithmic functions in a
slightly different way,

but calculus will once again be essential.

Although exponential and logarithmic functions define infinite classes of
functions, we are really

only interested in a single function from each of these classes. The exponential
function we will be

interested in is the base e exponential, where e is a specific irrational
number, defined by the limit

Because this function is used so much more often than other exponential
functions, it is often

referred to as the exponential function. Do not worry about understanding the
above notation -

as we delve into the study of limits and sequences it will begin to make sense.
Corresponding to

the base e exponential, we are interested in , the natural logarithm,
which is often written

as ln(x). The graphs of these two functions are given in figure 1.

Figure 1: Exponential and Logarithmic Functions

While the exponential function and natural logarithm are really the only two
functions of

interest for us, the base 10 logarithm is sometimes encountered as well. The
base 10 logarithm

often written log(x) for short, which was widely used to simplify calculations
before the advent of

computers. The base 10 logarithmic is a part of the definition of a decibel, so
it is encountered in

fields such as telecommunications and acoustics.