Algebra 2 and Trigonometry Honors Chapter 8: Logarithms Part A Name: Teacher: Pd: 1
Table of Contents Day 1: Inverses and Graphs of Logarithmic Functions & Converting an Exponential Equation into a Logarithmic Equation SWBAT: Convert an Exponential Equation into a Logarithmic Equation & Graph Logarithmic Functions Pgs. 3 9 in Packet HW: 10 11 in Packet Day 2: e and The Natural Log SWBAT: Learn the properties of e and the natural log Pgs. 12 17 in Packet HW: 18 21 in Packet Day 3: Properties of Logs SWBAT: Learn the properties of logs. Pgs. 22 26 in Packet HW: 27 28 in Packet Day 4: REVIEW DAY SWBAT: Review all the log properties Pgs. 29 32 in Packet HW: 29 32 in Packet QUIZ on Day 5 ***Answer Keys start at page 33**** 2
Day 1 Introduction into Logarithmic Functions Warm - Up: Using the table below: a) Complete the table of values for y= 2 x b) sketch the graph of y= 2 x x y -2-1 0 1 2 2) Recall: How do we find the inverse of a function? Find the inverse algebraically. 3) Graph the inverse of the function y = 2 x. Properties of Domain: Properties of Domain: Range: Range: Asymptote: Asymptote: x-intercept: x-intercept: y-intercept: y-intercept: 1
Concept 1: Example 1: Graph f(x) = ( 1 3 )x and its inverse f 1 (x). Ex 2: Graph y = log 5 x and its inverse and its inverse f 1 (x). 2
Regents Questions 3) 4) 5) The graph of the function y = 4 x appears in which two quadrants? (1) I and II (2) I and IV (3) II and III (4) III and IV 6) The graph of the function y = log 4 x appears in which two quadrants? (1) I and II (2) I and IV (3) II and III (4) III and IV 7) The inverse of the function y = log 25 x (1) y 25 = x (2) y = 25 x (3) x = 25 y (4) y = log x 25 8) The inverse of the function f(x) = 15 x (1) f 1 (x) = x 15 (2) f 1 (x) = log x 15 (3) f 1 (x) = log 15 x (4) f 1 (x) = log 15 y 3
Concept 2: Converting from Exponential to Logarithmic Form and Vice Versa Until now, there was no way to isolate y in an equation of the form x = 3 y. You cannot take the yth root of something if that something isn t a value. The word Logarithm means power. When you see the function log, you should translate that into the power I raise Example: Log 2 8 = 3 Example: X = Log 10 1000 The power I raise 2 to to get 8 is 3 The power I raise 10 to to get What is 10000 So, = 4
Concept 2: Simplifying a Log Expression Teacher Modeled What power?????" Student Try it! log 5 25 log 10 10, 000 log 4 1 log 5 625 log 6 ( 1 36 ) log 5 5 Regents Question Simplify a) Simplify: 7 log 7 4 b) Simplify: log 5 125 x 5
Concept 3: How do we change between log form and exponential form? Teacher Modeled Student Try it! log 8 x = 1 3 log 32 x = 2 5 log x 36 = 2 log x ( 216) = 3 log x 125 = 3 2 log x 243 = 5 2 6
Concept 4: How Do we evaluate Common Logs? Base with NO Exponent X Radical with NO index 25 means. means. Logarithms with a base log 5 x = 2 Logarithms with NO base log 1000 means. means. On the graphing Calculator 7
Evaluate each. Explanation log 100 1 log ( 10, 000 ) log 100 log 10 log 17 2log 4 8
Challenge: Solve the following Equation Summary/Closure: Exit Ticket: 9
Day 1 HW 10
27. Sketch below the graph of. a) State the domain and range of the graph. b) Write the equation of the asymptote. 28. 11
Day 2: e and the Natural Log Warm Up 1. 2. Fdf 3. 4. 12
There are many numbers in mathematics that are more important than others because they find so many uses in either mathematics or science. Good examples of important numbers are 0, 1, i, and. In this lesson you will be introduced to an important number given the letter e for its inventor Leonhard Euler (1707-1783). This number plays a crucial role in Calculus and more generally in modeling exponential phenomena. THE NUMBER e 1. Like, e is irrational. 2. e 3. Used in Exponential Modeling Exercise #1: Which of the graphs below shows y e x? Explain your choice. Check on your calculator. (1) y (2) y (3) y (4) y x x x x Explanation: Because of the importance of y e x, its inverse, known as the natural logarithm, is also important. The natural logarithm, like all logarithms, gives an exponent as its output. In fact, it gives the power that we must raise e to in order to get the input. Exercise #2: Without the use of your calculator, determine the values of each of the following. (a) ln e (b) THE NATURAL LOGARITHM The inverse of : 5 ln 1 (c) ln e (d) ln e 13
Exercise #3: Exercise # 4: x Exercise #5: On the grid below, the solid curve represents y e. Which of the following exponential functions could describe the dashed curve? Explain your choice. y x (1) y 1 2 (3) y 2 x (2) y e x (4) y 4 x x Exercise #6: Determine the domain of the function y log 3x 4 notation.. State your answer in set-builder 2 14
Exercise #7: Identify the domain and range of each. Then sketch the graph. 15
Exercise #8: A hot liquid is cooling in a room whose temperature is constant. Its temperature can be modeled using the exponential function shown below. The temperature, T, is in degrees Fahrenheit and is a function of the number of minutes, m, it has been cooling. 0.03 e T m m 101 67 (a) What was the initial temperature of the water at m 0. Do without using your calculator. (b) How do you interpret the statement that T 60 83.7? (c) Using the natural logarithm, determine algebraically when the temperature of the liquid will reach 100 F. Show the steps in your solution. Round to the nearest tenth of a minute. (d) On average, how many degrees are lost per minute over the interval 10 m 30? Round to the nearest tenth of a degree. 16
Summary Exit ticket 17
1. The domain of y log x 5 3 Day 2 HW in the real numbers is (1) x x 0 (3) x x 5 (2) x x 5 (4) x x 4 2. Which of the following equations describes the graph shown below? (1) y log5 x (3) y log3 x y (2) y log 2 x (4) y log 4 x x 3. Which of the following represents the y-intercept of the function y x (1) 8 (3) 1 (2) 4 (4) 4 log 32 1? 4. Which of the following values of x is not in the domain of f x log 10 2x (1) 3 (3) 5 (2) 0 (4) 4 5. Which of the following is true about the function y x (1) It has an x-intercept of 4 and a y-intercept of 1. (2) It has x-intercept of 12 and a y-intercept of 1. (3) It has an x-intercept of 16 and a y-intercept of 1. (4) It has an x-intercept of 16 and a y-intercept of 1. log 16 1? 4 5 2? 18
6. Determine the domains of each of the following logarithmic functions. State your answers using any accepted notation. Be sure to show the inequality that you are solving to find the domain and the work you use to solve the inequality. (a) y log 2x 1 (b) y log 6 x 5 7. Which of the following is closest to the y-intercept of the function whose equation is (1) 10 (3) 27 (2) 18 (4) 52 y 1 10e x? x 8. On the grid below, the solid curve represents y e. Which of the following exponential functions could describe the dashed curve? Explain your choice. y x (1) y 1 2 (3) y 2 x (2) y e x (4) y 4 x x 9. Which of the following values of t solves the equation (1) ln15 10 (2) 1 2ln 5 (3) 2ln3 (4) ln 3 2 2t 5 15 e? f x 10. At which of the following values of x does 2 x 2e 32 have a zero? (1) 5 ln 2 (3) ln8 (2) ln 4 (4) 2 y ln 5 19
ct 13. For the equation ae d, solve for the variable t in terms of a, c, and d. Express your answer in terms of the natural logarithm. 20
APPLICATIONS 14. Flu is spreading exponentially at a school. The number of new flu patients can be modeled using the 0.12d equation F 10e, where d represents the number of days since 10 students had the flu. (a) How many days will it take for the number of new flu patients to equal 50? Determine your answer algebraically using the natural logarithm. Round your answer to the nearest day. (b) Find the average rate of change of F over the first three weeks, i.e. 0 d 21. Show the calculation that leads to your answer. Give proper units and round your answer to the nearest tenth. What is the physical interpretation of your answer? 21
Day 3 Properties of Logarithms 22
23
24
25
Challenge Summary/Closure: Exit Ticket: 26
Day 3 - HW 27
28
Day 4: Review of Logarithms (Days 1 3) 29
30
Common Core Problems Sets 31
32
Answer Keys Day 1 27. y = log x log 4 28. x y 0 error 1 0 2.5 4 1 8 1.5 33
Day 2 7. 8. 34
7. 8. 9. 10. 11. C 12. A 13. 35
14. 36
Day 3 37
38
Day 4: Common Core Problem Sets 39
40
41