Part 4: Exponential and Logarithmic Functions

Transcription

1 Part 4: Exponential and Logarithmic Functions Chapter 5 I. Exponential Functions (5.1) II. The Natural Exponential Function (5.2) III. Logarithmic Functions (5.3) IV. Properties of Logarithms (5.4) V. Solving Exponential and Log Equations (5.5) VI. Solving Exponential and Log Inequalities (5.5) VII. Compound Interest (5.6) VIII. Exponential Growth and Decay (5.7) Note: Sections I IV are review material. Part 4 1

2 I. Exponential Functions 5.1 Recall: Some of the properties of exponents: (1) Product of powers (2) Power of powers Example: Use these properties to simplify the following expressions (1) (3) Power of a product (2) (4) Quotient of powers Part 4 2

3 Definition: The exponential function with base b is defined to be the equation Examples (Determine which are exponential functions) HOW DO YOU DETERMINE IF YOU HAVE AN EXPONENTIAL FUNCTION? Part 4 3

4 Graphing an exponential function Graph x y y x Determine the following: Domain y intercept Asymptotes Range x intercept Part 4 4

5 Recall from Part 2: Transformation of y = f(x) Overview Part 4 5

6 Ex: Based on the graph of 1) 4), graph the following. 2) 3) 5) Part 4 6

7 Compare the Properties of the graphs y = 2 x and graph domain range intercepts asymptotes end behaviour Q. Why do the two graphs differ? Part 4 7

8 Generalization of y y x x Domain = Range = y intercept = x intercept = Asymptotes: Part 4 8

9 Ex: Find an exponential function of the form f(x) = Ca x whose graph is given. ( 1, 15) 5 Part 4 9

10 ` II. The Natural Exponential Function 5.2 Given the function, complete the table of values. "Use all decimal places calculator allows" x f(x) ,000 10, ,000 1,000,000 Part 4 10

11 The Natural Exponential Function: Ex: Graph Domain = e y Range = y intercept = x intercept = Asymptotes: x Remark: Like the number π, e is an irrational number. The number e has significance in many areas & has a strong importance in several applications. Part 4 11

12 Later we will discuss solving exponential functions more generally, however, we can solve the case where we have common bases. Ex: Solve Part 4 12

13 III. Logarithmic Functions 5.3 Recall how we graphed How could we graph the inverse? Graph with its inverse & y = x. function x = inverse x y x y y x The exponential function has a inverse since it is a function! Part 4 13

14 Logarithmic Functions Def: (Alternate way to write definition) means where You should be able to change from log form to exponential form and vice versa! Ex: Complete the following Exponential Form Logarithmic Form Part 4 14

15 Ex: Evaluate Part 4 15

16 Properties: 1. WHY? Part 4 16

17 Ex: Solve the log equations Part 4 17

18 Let's look at the restrictions for exponential and log functions.. Restrictions on base b Domain: Range: Part 4 18

19 Conclusion: Domain of is You cannot take the logarithm of Example. Find the domain of the log function Part 4 19

20 Ex: Find the domain of Part 4 20

21 Graphs of log functions exponential form: exponential form: y y x 1 x Q. What do you notice? Part 4 21

22 y Domain = Range = y x Asymptote: x intercept = x Recall: y Domain = Range = y x Asymptote: y intercept = x Part 4 22

23 Ex: Graph Domain = Range = y intercept = x intercept = Asymptote: y x Part 4 23

24 Ex: Graph Domain = Range = y intercept = x intercept = Asymptote: y x Part 4 24

25 Two Special Logs What makes these logs special?? Ex: Use calculator to estimate the following to 4 decimal places. 1) log 7 2) ln7 3) log.23 4) log( 2) Part 4 25

26 IV. More Properties of Logs 5.4 Let P & Q be positive real numbers. Log Property Corresponding Exponent Property 1) 2) 3) for all real numbers r. Part 4 26

27 Proof of Property 1 Property 1: Proof: Let x = log b P and let y = log b Q Then P = & Q = Part 4 27

28 Ex: Write the following as a single logarithm. 1) 2) 3) Part 4 28

29 Ex: Using Log Properties to Expand Expressions 1) 2) Part 4 29

30 Ex: Write the following as a single logarithm. Part 4 30

31 Ex: Write the following as a single logarithm. Part 4 31

32 Ex: Using Log Properties to Expand Expressions Part 4 32

33 Ex: Using Log Properties to Expand Expressions Part 4 33

34 Ex: Using Log Properties to Expand Expressions Part 4 34

35 Example: Given that log b 2 = A and log b 6 = B, express each in terms of A and/or B Part 4 35

36 Previously, we learned how to solve exponential equations if they had a common base. What happens if we cannot make a common base??? Example: Solve Hint: Re write it in log form!! Part 4 36

37 Change of Base Theorem: What is a? In particular, What is important about these two forms?? Part 4 37

38 We could also solve by using the laws of logs. Specifically, we can apply Part 4 38

39 V. Solving Exponential and Logarithm Equations 5.5 Solve for x Note: Know the difference between an exact answer and an approximate answer. Part 4 39

40 Ex: Graph Domain = Range = y intercept = x intercept = Asymptote: y x Part 4 40

41 Ex: Graph Domain = Range = y intercept = x intercept = Asymptote: y x Part 4 41

42 Find the inverse function Part 4 42

43 Ex: Find the real roots. Write both the exact and approximate answer. (1) Part 4 43

44 (2) Part 4 44

45 Solve these equations. Find both the exact and approximate solutions. (1) (2) (3) Part 4 45

46 Solve the following equations. Find the exact and approximate answers. Part 4 46

47 Solve for x: Part 4 47

48 Solve for x: Part 4 48

49 Solve. Part 4 49

50 Example: Find the intercepts. Part 4 50

51 VI. Solving Exponential and Logarithm Inequalities!! The following are true for b > ) Refer to figure 1 2) Refer to figure 2 Figure 1 b q y = b x, b > 1 log b q y = log b x, b>1 p b p q log b p p q Figure 2 What do these properties mean??? (1) Part 4 51

52 Solve the inequality. Steps: (1) What type of inequality is it? (2) Isolate. (3) Solve appropriately. Part 4 52

53 Solve the inequality. Part 4 53

54 Recall: Example: Find the domain of Part 4 54

55 Solve the inequality. Steps to solving this log inequality: (1) Find the domain of the log part(s). (2) Isolate the log part. (3) How do you get rid of the log? (4) Simplify and solve. (5) Why did I find the domain??? Part 4 55

56 Solve the inequality. Part 4 56

57 Practice with logs. (1) Solve. Part 4 57

58 (2) Solve. Part 4 58

59 VII. Compound Interest 5.6 Where do we use exponential and log equations in real life??? Q: Why is the compound interest formula considered an exponential equation??? Part 4 59

60 What is the difference between simple interest and compound interest?? Simple interest at 10% annually. January December Compound interest at 10% annually compounded twice a year. January December Part 4 60

61 Use a calculator to answer the following. Suppose $5000 is invested into a savings account with an annual interest rate of 6%. Find the amount in the savings account after 6 years given: (1) Compounded annually (2) Compounded quarterly Part 4 61

62 Ex. Suppose that $2000 is invested at 7.5% annual interest compounded annually. How many years will it take for the money to double? Part 4 62

63 Compound Interest Continuously Ex: Suppose $5000 is invested into a savings account with an annual interest rate of 6%. Find the amount in the savings account after 6 years given that the account compounds continuously. Part 4 63

64 Ex: If Alana opens a savings account with initial deposit of $1000 that is compounded monthly with interest 8% per year, how long will it take 1) To have a balance of $ ) To double her investment. Part 4 64

65 Ex: If Alana opens a savings account with initial deposit of $1000 that is compounded continously with interest 8% per year, how long will it take 1) To have a balance of $ ) To double her investment. Part 4 65

66 VIII. Exponential Growth / Decay 5.7 In biology, economics, and social sciences we have applications where a quantity changes at a rate proportional to the amount present. These situations produce an exponential function. where n(t) = quantitiy (or population) at time t n 0 = initial quantity (or initial population) n(0) = In the applications, r must be determined. The application models growth if r > 0 and decay if r < 0. Graph of n(t) n(t) n o n o t t r > 0 (Growth) r< 0 (Decay) Part 4 66

67 Ex: A population of bacteria in a culture is increasing exponentially. The original culture of 25,000 bacteria contains 40,000 bacteria after 10 hours. How long will it be until there are 60,000 bacteria in the culture. Part 4 67

68 Ex: If 600 grams of a radioactive substance are present initially and 3 years later only 300 grams remain, how much of the substance will be present after 6 years? Part 4 68

69 Carbon Dating: Carbon 14 is a radioactive form of carbon found in all living plants and animals. After a plant or animal dies, carbon 14 decays exponentially with a half life of 5600 years. The quantity Q of Carbon 14 remaining is given by the equation Ex: Find the rate r in carbon 14 dating. Round to six decimal places. Part 4 69

70 Ex: The Lascaux caves of France contain prehistoric paintings of animals. Charcoal found in these caves contains 15% of the amount of carbon 14 in living trees. Approximate the age of the paintings. (You will need results found from previous problem) Part 4 70

71 Part 4 71