Transcript Exponential Functions

5
Exponential and Logarithmic Functions
Exponential and Logarithmic Functions
5.1
Exponents and Exponential Functions
Objectives
• Review the laws of exponents.
• Solve exponential equations.
• Graph exponential functions.
Exponents
Property 5.1
If a and b are positive real numbers and m and n
are any real numbers, then the following properties
hold:
n
m
n m
b

b

b
1.
Product of two powers
n m
mn
(
b
)

b
2.
n
n n
3. (ab)  a b
n
n
a
a


4.

b
 
Power of a power
Power of a product
Power of a quotient
bn
n
b
n m
5.

b
bm
Quotient of two powers
Exponents
Property 5.2
If b > 0 but b  1, and if m and n are real numbers,
then
bn = bm if and only if n = m
Exponents
Solve 2x = 32.
Example 1
Exponents
Solution:
2x = 32
2x = 25
32 = 25
x=5
Apply Property 5.2
The solution set is {5}.
Example 1
Exponential Functions
If b is any positive number, then the expression bx
designates exactly one real number for every real
value of x. Therefore the equation f(x) = bx defines
a function whose domain is the set of real numbers.
Furthermore, if we add the restriction b  1, then
any equation of the form f(x) = bx describes what
we will call later a one-to-one function and is called
an exponential function.
Exponential Functions
Definition 5.1
If b > 0 and b  1, then the function f defined by
f (x) = bx
where x is any real number, is called the exponential
function with base b.
Exponential Functions
• The function f (x) = 1x is a constant function
(its graph is a horizontal line), and therefore
it is not an exponential function.
Exponential Functions
Graph the function f (x) = 2x.
Example 6
Exponential Functions
Example 6
Solution:
Let’s set up a table of values. Keep in mind that the domain is
the set of real numbers and the equation f (x) = 2x exhibits no
symmetry. We can plot the points and connect them with a
smooth curve to produce Figure 5.1.
Figure 5.1
Exponential Functions
The graphs in Figures 5.1 and 5.2 illustrate a general
behavior pattern of exponential functions. That is, if b > 1,
then the graph of f (x) = bx goes up to the right, and the
function is called an increasing function. If 0 < b < 1,
then the graph of f (x) = bx goes down to the right, and the
function is called a decreasing function.
Continued . . .
Figure 5.2
Exponential Functions
These facts are illustrated in Figure 5.3. Notice that b0 = 1 for
any b > 0; thus, all graphs of f (x) = bx contain the point
(0, 1). Note that the x axis is a horizontal asymptote of the
graphs of f (x) = bx.
Figure 5.3