Section 4.2 - Exponential Functions

Section Objectives

  1. Evaluate exponential functions.
  2. Graph exponential functions.
  3. Use exponential functions in applications.

Exponential Functions

Let be a fixed positive real number not equal to 1. The base- exponential function is defined by .


Properties of the Exponential Functions

In general, the exponential functions have the following properties.


Solving Elementary Exponential Equations

Equations involving exponential functions arise in many applications. Certain kinds of exponential equations, namely those involving exponentials of the same base, have a very simple solution method. The method is based on the fact that if and only if .

In order to use this idea to solve an exponential equation,

  1. Rewrite the equation so that a single term is on each side.
  2. Write each side in terms of the same base.
  3. Write the new equation that says the exponents are equal.
  4. Solve the new equation.


The Base- Exponential

The number is a special mathematical constant that arises in many places in mathematics. This number is defined by the following limit: as .

is an irrational number whose value is approximately .

is very often used as a base for an exponential function: . In fact, is usually called the "natural base".


Applications - Population Growth

A very common population growth model is to assume that a population grows (or decays) according to a formula of the form , where , the initial population at time , and the base is a number the characterizes the rate of growth (or decay). (In fact, is the growth rate per unit time.)


VERY IMPORTANT IDEA: The model describes growth if , and it describes decay if .

Applications - Compound Interest

An investment of dollars at an annual interest rate of compounded times per year has an accumulated value after years of .

If the interest is compounded continuously, .


Applications - Radioactive Decay

Radioactive decay is modeled by a formula identical to that of exponential population growth, except that the base is between 0 and 1: , , where is the initial amount of the radioactive substance.

The half-life of a radioactive isotope is the amount of time required for half of a given amount to decay. The half-life, , satisfies the equation .

If you know the half-life, this equation can be used to find the base .