Functions and Models: Exponential Functions
base, exponent, exponential function, rational number, irrational number, transcendental number, successive approximation, rate of growth, half-life
Defining exponential functions
The exponential function can be algebraically defined when is rational. When is irrational, we define as a limit of powers of where the exponent is a sequence of rational numbers converging to .
Positive Integer Exponents
If is a positive integer, then we can think of as a shorthand notation for multiplying 2 times itself times.
Example: and similarly .
Negative Integer Exponents
If is a negative integer, then we define as the reciprocal of .
Example: and .
If is a rational number, then we define as the th root of .
For this example, we take the base of the exponential function to be . Let us also take a particular case of an irrational exponent for purposes of illustration. Take the exponent . Using Maple, we can find decimal approximations of to any desired number of digits. These approximations up to 3 decimal digits are as follows:
We can use these rational approximations to to obtain a sequence of approximations to (since we are raising 2 to a rational power each time, each of these powers is defined as in the previous section).
As we use more decimal digits in our rational approximations of , our approximations of become progressively better. The limit of these approximations is the value of .
Laws of Exponents
We record the following properties of exponential calculations. These properties will be useful in simplifying computations in later problems.
In previous algebra courses, you have likely proved the laws of exponents for rational values of . These laws in fact apply to all real values of . We will prove these laws later in the course.
Modelling with Exponential Functions
Exponential functions are useful in modeling many phenomena involving rapid growth. In particular, population growth under ideal conditions can be modeled effectively by exponential functions. Let represent the size of a population at time . The basic exponential growth model is given by the following equation where is the doubling time and is the initial population:
Example: If a bacteria colony initially contains 100 bacteria and the population size doubles every 3 hours, what is the size of the population after 10 hours?
The size of the population after 10 hours is , which is approximately 1007 bacteria.
Exponential functions are useful in modeling radioactive decay. Let represent the amount of a radioactive substance present at time . The basic exponential decay model is given by the following equation where is the half-life and is the initial amount of the radioactive substance:
Example: If there are initially 50 mg of the radioactive isotope Strontium-90 and the half-life of this isotope is 25 years, how much of the isotope remains after 13 years?
The amount of the isotope after 13 years is milligrams, which is approximately 34.869 mg.
The Natural Base
Later in this course we will want to simplify our computations involving exponential functions. We introduce a special constant now which serves this purpose. Among all the possible values of the base , there is only one value for which the graph of has slope 1 at the point (0,1). This value of is the number , named after the mathematician Euler who also showed that is irrational (approximately a century later, Hermite showed that is in fact transcendental). The first digits of the decimal representation of are 2.718281828. In later sections we will introduce other ways of defining .