Exponential and Logarithmic Functions 5

5.1 Exponents 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

Exponents Property 5.2 If b > 0 but b 1, and if m and n are real numbers, then b n = b m if and only if n = m

Exponents Solve 2 x = 32. Example 1

Exponents Solution: 2 x = 32 2 x = = 2 5 x = 5 Apply Property 5.2 The solution set is {5}. Example 1

Exponential Functions If b is any positive number, then the expression b x designates exactly one real number for every real value of x. Therefore the equation f(x) = b x 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) = b x 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) = b x where x is any real number, is called the exponential function with base b.

Exponential Functions The function f (x) = 1 x 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) = 2 x. 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) = 2 x 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) = b x goes up to the right, and the function is called an increasing function. If 0 < b < 1, then the graph of f (x) = b x 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 b 0 = 1 for any b > 0; thus, all graphs of f (x) = b x contain the point (0, 1). Note that the x axis is a horizontal asymptote of the graphs of f (x) = b x. Figure 5.3