ExponentialandLogarithmicFunctions.ppt

3 Exponential Functions Logarithmic Functions Exponential Functions as Mathematical Models 3.1 Exponential Functions Exponential Function The function defined by is called an exponential function with base b and exponent x. The domain of f is the set of all real numbers. Example The exponential function with base 2 is the function with domain (– ?, ?). The values of f(x) for selected values of x follow: Example The exponential function with base 2 is the function with domain (– ?, ?). The values of f(x) for selected values of x follow: Laws of Exponents Let a and b be positive numbers and let x and y be real numbers. Then, Examples Let f(x) = 22x – 1. Find the value of x for which f(x) = 16. Solution We want to solve the equation 22x – 1 = 16 = 24 But this equation holds if and only if 2x – 1 = 4 giving x = . Examples Sketch the graph of the exponential function f(x) = 2x. Solution First, recall that the domain of this function is the set of real numbers. Next, putting x = 0 gives y = 20 = 1, which is the y-intercept. (There is no x-intercept, since there is no value of x for which y = 0) Examples Sketch the graph of the exponential function f(x) = 2x. Solution Now, consider a few values for x: Note that 2x approaches zero as x decreases without bound: There is a horizontal asymptote at y = 0. Furthermore, 2x increases without bound when x increases without bound. Thus, the range of f is the interval (0, ?). Examples Sketch the graph of the exponential function f(x) = 2x. Solution Finally, sketch the graph: Examples Sketch the graph of the exponential function f(x) = (1/2)x. Solution First, recall again that the domain of this function is the set of real numbers. Next, putting x = 0 gives y = (1/2)0 = 1, which is the y-intercept. (There is no x-intercept, since there is no value of x for which y = 0) Examples Sketch the graph of the exponential function f(x) = (1/2)x. Solution Now,