Exponential and Logarithmic Functions

5

5.1Exponents and Exponential Functions

Exponential and Logarithmic 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:

1. Product of two powers

2. Power of a power

3. Power of a product

4. Power of a quotient

5. Quotient of two powers

n m n mb b b ( )n m mnb b

( )n n nab a bn n

n

a a

b b

nn m

m

bb

b

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 byf (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

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.

Example 6

Figure 5.1

Exponential Functions

Figure 5.2

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 . . .

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