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Transcript
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