Chapter 10: Exponential and Logarithmic Functions

1. Chapter 10: Exponential and Logarithmic Functions

2. 10.1 Algebra and Composition of Functions

3. Sum, Difference, Product, and Quotient of Functions Given two functions f and g, the functionsf + g, f – g, f ∙ g and are defined as The domain of each new function is the intersection of the domains of the original functions.

4. Examples Given the functions f(x) = 10x + 7 and g(x) = x2 – 5x, find the following functions and their domains. • (f + g)(x) = f(x) + g(x) = (10x + 7) + (x2 – 5x) = x2 + 5x + 7 • (f – g)(x) = f(x) – g(x) = (10x + 7) – (x2 – 5x) = –x2 + 15x + 7 The domain of each of these functions is the intersection of the domains of f and g, which are all real numbers. So the domain of each new function is (–∞, ∞).

5. Composition of Functions The composition of f and g, denoted f◦ g, is defined by the rule (f◦ g)(x) = f(g(x)) provided g(x) is in the domain of f. The composition of g and f, denoted g◦ f, is defined by the rule (g◦ f)(x) = g(f(x)) provided f(x) is in the domain of g.

6. In the composition (f◦ g)(x) = f(g(x)) , the function g is the innermost operation and acts on x first. Then the output of function g becomes the domain element of f.

7. The order in which two functions are composed may result in different functions. f(x) = x – 3 g(x) = x2 + 4 (f◦ g)(x) = f(g(x)) = f(x2 + 4) = (x2 + 4) – 3 = x2 + 1 (g◦ f)(x) = g(f(x)) = g(x – 3) =(x – 3)2 + 4 = x2 – 6x + 9 + 4 = x2 – 6x + 13

8. Exercise 1 If f(x) = 13x + 7 and g(x) = x2 – 3x, find (f – g)(x). A) 13x2 – 39x + 7 B) x2 + 10x + 7 C) 13x3 – 32x2 – 21x D) –x2 + 16x + 7

9. Exercise 2 If f(x) = 10x + 7 and g(x) = x2 – 4x, find (f∙ g)(x). A) 10x2 – 40x + 7 B) x2 + 6x + 7 C) 10x3 – 33x2 –28x D) –x2 + 14x + 7

10. Exercise 3 When f(x) = x + 4 and g(x) = 5x2 + 6x, find (f ◦ g)(x). Write the domain in interval notation.    A) (f ◦ g)(x) = 5x2 + 6x + 4; (–∞, ∞) B) (f ◦ g)(x) = 5x2 + 6x + 4; [–∞, ∞] C) (f ◦ g)(x) = 5x2 + 7x + 4; (–∞, ∞)  D) None of the above

11. 10.2 Inverse Functions

12. A function f is a set of ordered pairs (x, y) such that for every element x of the domain there corresponds exactly one element y in the range. • The inverse of f, f–1, interchanges the values of x and y so that the domain of f is the same as the range of f–1, and the range of f is the domain of f–1. • f–1 denotes the inverse of a function. The –1 does not represent an exponent.

13. A necessary condition for a function f to have an inverse function is that no two ordered pairs in f have different x-coordinates and the same y-coordinate. • A function that satisfies this condition is called a one-to-onefunction. • The vertical line test is used to determine visually if a graph represents a function. The horizontal line test is used to determine whether a function is one-to-one.

14. Horizontal Line Test Consider a function defined by a set of points (x, y) in a rectangular coordinate system. The graph of the ordered pairs defines y as a one-to-one function of x if no horizontal line intersects the graph in more than one point. one-to-one not one-to-one

15. Finding an Equation of an Inverse of a Function For a one-to-one function defined by y = f(x), the equation of the inverse can be found as follows: • Replace f(x) by y. • Interchange x and y. • Solve for y. • Replace y by f–1(x).

16. Example Find the inverse of the one-to-one function.f(x) = 4x3 – 3

17. The graphs of a function and its inverse are symmetric with respect to the line y = x.

18. For a function that is not one-to-one, sometimes its domain can be restricted to create a new function that is one-to-one. For example, y= x2 + 4 is a parabola and not one-to-one, but if we restrict the domain to x> 0 the graph consists only of the “right” branch of the parabola, which is a one-to-one function.

19. Definition of the Inverse of a Function If f is a one-to-one function, then g is the inverse of f if and only if (f◦g)(x) = x for all x in the domain of g and (g◦f)(x) = x for all x in the domain of f

20. Example Show that the functions are inverses.

21. Exercise 4 Using the horizontal line test, is the following function a one-to-one function? A) No; the line x = –1 intersects the curve at more than one point. B) No; the line y = –1 intersects the curve at more than one point. C) No; the line y = x intersects the curve at more than one point. D) Yes

22. Exercise 5 Find the inverse of the function f(x) = 6x – 8.   A) B) C) D)

23. Exercise 6 If f(x) = x3 – 5 and , then   A) g is the inverse function of f B) f is the inverse function of g C) (f◦g)(x) = x and (g◦f)(x) = x D) All of the above are true.

24. 10.3 Exponential Functions

25. Definition of an Exponential Function Let b be any real number such that b > 0 and b≠ 1. Then for any real number x, a function of the form y = bx is called an exponential function. Its domain is all real numbers. An exponential function has a constant base and a variable exponent. y = 3x y = 7x y = (0.5)x

26. Graphs of f(x) = bx The graph of an exponential function defined by f(x) = bx (b > 0 and b≠ 1) has the following properties: • If b > 1, f is an increasing exponential function, sometimes called an exponential growth function.If 0 < b < 1, f is a decreasing exponential function, sometimes called an exponential decay function. • The domain is all real numbers, (–,  ). • The range is (0, ). • The x-axis is a horizontal asymptote. • The function passes through the point (0, 1) because f(0) = b0 = 1.

27. Graphs of Exponential Functions • Domain: (–,  ) • Range: (0, ) • x-axis is horizontal asymptote • Graphs pass through (0, 1)

28. Applications of Exponential Functions Exponential growth and decay can be found in a variety of real-world phenomena. For example: • Population growth can often be modeled by an exponential function. • The growth of an investment under compound interest increases exponentially. • The mass of a radioactive substance decreases exponentially with time.The half-life of a radioactive substance is the amount of time it takes for one-half the original amount of the substance to change into something else.

29. Exercise 7 Which is the graph of ? A) B) C) D)

30. Exercise 8 Which is the graph of f(x) = 4–x? A) B) C) D)

31. Exercise 9 The population of creepy crawly gross things in a dorm refrigerator was 2000 at noon, and was increasing at a rate of 10% per hour. The number can be found using the function where t is the number of hours past noon. Predict the number of creepy crawly gross things at 11 PM. P(t) = 2000(1.1)t A) 3706 B) 24,200 C) 5706 D) 7726

32. 10.4 Logarithmic Functions

33. Definition of a Logarithm Function If x and b are positive real numbers such that b≠ 1, then y = logb(x) is called the logarithmic function with base b and y = logb(x) is equivalent to by = x. In the expression y = logb(x), y is called the logarithm,b is called the base, and x is called the argument.

34. The expression y = logb(x) indicates that the logarithm y is the exponent to which b must be raised to obtain x. • The expression y = logb(x) is called the logarithm form of the equation. • The expression by = x is called the exponential form of the equation.

35. Examples Evaluate the logarithmic expressions. • log101000 is the exponent to which 10 must be raised to obtain 1000.y = log10100010y = 1000y = 3 Therefore log101000 = 3. • log3243 is the exponent to which 3 must be raised to obtain 243.y = log32433y = 243y = 5 Therefore log3243 = 5.

36. The Common Logarithmic Function The logarithmic function with base 10 is called the common logarithmic function and is denoted by y = log(x). The base is not explicitly written but is understood to be 10. Most calculators have a log key to compute logarithms with base 10.

37. Graphs of Logarithmic Functions f(x) = logbx is the inverse of g(x) = bx, so their graphs are symmetric about the line y = x. The domain of y = logbx is the set of positive real numbers. The range is (–,), and the y-axisis a vertical asymptote.

38. Graphs of Exponential and Logarithmic Functions: Summary

39. Exercise 10 Convert to logarithmic form. 43 = 64 A) log4 3 = 64 B) log4 64 = 3 C) log64 4 = 3 D) log3 4 = 64

40. Exercise 11 Find the domain of the function. Write your answer in interval notation. f(x) = log(2x + 10) A) (0, ) B) (5, ) C) (–5, ) D) [–5, 5]

41. Exercise 12 The level of a sound in decibels is calculated using the formula where I is the intensity of the sound waves in watts per square meter. A jackhammer puts out 1 watt per square meter. How many decibels is that? A) 12 B) 120 C) 1200 D) 10

42. 10.5 Properties of Logarithms

43. Properties of Logarithms From the definition of logarithms,y = logb (x) is equivalent to by = x for x > 0, b > 0, and b ≠ 1. The following properties follow directly from the definition. Property 1 logb 1 = 0 Property 2 logb b = 1 Property 3 logb (bp) = p Property 4

44. Product Property for Logarithms Let b, x, and y be positive real numbers where b≠ 1. Then logb (xy) = logb x + logb y The logarithm of a product equals the sum of the logarithms of the factors.

45. Quotient Property for Logarithms Let b, x, and y be positive real numbers where b≠ 1. Then The logarithm of a quotient equals the difference of the logarithms of the numerator and denominator.

46. Power Property for Logarithms Let b and x be positive real numbers where b≠ 1. Let p be any real number. Then logb (xp) = p logb x

47. Example Use the properties of logarithms to write the expression as the sum or difference of logarithms. Assume that all variables represent positive real numbers.

48. Example Use the properties of logarithms to rewrite the expression as a single logarithm. Assume that all variables represent positive real numbers. 2 log x + 13 log y = log x2 + log y13Power property of logarithms = log (x2y13) Product property for logarithms

49. Exercise 13 Write as a sum or difference of logarithms of a, b and c. Assume that all variables represent positive real numbers. log1/2(abc)   A) (log1/2a)(log1/2b)(log1/2c) B) log1/2a + log1/2b + log1/2c C) log1/2a – log1/2b – log1/2c D)

50. Exercise 14 Write as a single logarithm and simplify, if possible. Assume that all variables represent positive numbers. log5 2 + log5 100 – log5 8 A) 2 B) log5 94 C) D) log5 1600