Pre-CalculusChapter 3

Exponential and Logarithmic Functions

3.1 Exponential FunctionsObjectives: Recognize and evaluate exponential

functions with base a. Graph exponential functions. Recognize, evaluate, and graph

exponential functions with base e. Use exponential functions to model and

solve real-life problems.

Definition of Exponential Function

The exponential function with base a is denoted by

where a > 0, a ≠ 1, and x is any real number.

xaxf )(

Properties of Exponents

1.4

11.3

.2

.1

0

a

aaa

aa

a

aaa

x

xx

yxy

x

yxyx

222.8

.7

.6

.5

aaa

b

a

b

a

aa

baab

x

xx

xyyx

xxx

Exponential Function The exponential function

is different from the other functions we have studied so far.

The variable x is an exponent. A distinguishing characteristic of an

exponential function is its rapid increase as x increases (for a > 1).

Characteristics of f (x) = ax

Characteristics of f (x) = a -x

Transformations of f (x) = ax

The graph of g(x) = a(x ± h) is a ______ shift of f.

The graph h(x) = ax ± k is a ________ shift of f.(Note new horizontal asymptote.)

The graph k(x) = –ax is a reflection of f ______.

The graph j(x) = a(–x) is a reflection of f ______.

The Natural Base, e The natural base (or Euler’s

Number) is a constant that occurs frequently in nature and in science.

e ≈ 2.718281828459 Like π, e is irrational. It continues

forever in a non-repeating manner. The natural base function is f (x) = ex. The properties and characteristics of

exponential functions hold for ex.

Graph of ex

Complete the table and graph the function.

x ex

-2

-1

0

1

2

                       

                       

                       

                       

                       

                       

                       

                       

                       

                       

                       

Applications Exponential functions are used to

model rapid growth or decay. Examples include:

Compound interest Population growth Radioactive decay

Compound Interest

Example 1 A total of $12,000 is invested at an

annual rate of 3%. Find the balance after 4 years if the interest is compounded

a. Quarterlyb. Continuously

Example 2 Let y represent a mass of radioactive

strontium (90Sr), in grams, whose half-life is 28 years. The quantity of strontium present after t years is

a. What is the initial mass (when t = 0)?

b. How much of the initial mass is present after 80 years?

28

2

110

t

y

Example 3 The approximate number of fruit

flies in an experimental population after t hours is given by where t ≥ 0.

a. Find the initial number of fruit flies in the population.

b. How large is the population of fruit flies after 72 hours?

tetQ 03.020)(

Homework 3.1 Worksheet 3.1