PROBLEM SOLVING, REASONING AND LOGARITHMS INVERSE FUNCTIONS 1. So far, you have learned a lot about eight different parent graphs: y = x2 y = x3 y=x y= x y= x y = 1/x y = bx x 2 + y2 = 1 a. For each parent, find the inverse. Be sure to write the equation of the inverse in y = form. Include a sketch of each Parent Graph and its inverse. Remember that you can use DrawInv on your graphing calculator to help test your ideas. b. Are any parent functions their own inverses? Explain how you know. c. Do any parent functions have inverses that are not functions? If so, which ones? 2. Two parent functions, y = x and y = 2 x , have inverses for which you do not know how to write an equation in “ y = ” form. You will come back to y = x later. Since, as you know, exponential functions are so useful for modeling, it would make sense that the inverse of an exponential function is also important. Even though you may not know how to write the inverse of y = 2 x in “ y = ” form, you do know a lot about it: a. You know how to make an x → y table for the inverse of y = 2 x . Make the table. b. You also know what the graph looks like. Sketch a graph of the inverse. c. You also know what an equation for this table and graph looks like. Write an equation for the inverse, even though it will not be in “ y = ” form. 3. SILENT BOARD GAME: Your teacher will put an incomplete x → y table on the board or overhead. Fill in the missing values in the table. If you get stuck try the ancient puzzle in the next problem. x 8 g(x) 3 32 1/2 1 -1 16 4 3 64 6 2 0 0.25 -1 2 0.2 1/8 AN ANCIENT PUZZLE 4. Below is a puzzle that is more than 2100 years old. It was first conceived by mathematicians in ancient India in the 2nd century BCE. More recently, about 700 years ago, Muslim mathematicians created the first tables allowing them to find answers to this puzzle quickly. Tables similar to them appeared in school math books until recently. Here are some clues to help you figure out how the puzzle works: log2 4 = 2 log3 27 = 3 log5 25 = 2 log10 10,000 = 4 Use the clues to find the missing pieces of the puzzles below: a. log2 8 = ? b. log2 32 = ? c. log? 100 = 2 d. log5? = 3 e. log? 81 = 4 f. log100 10 = ? 5. While the idea behind the Ancient Puzzle is more than 2100 years old, the symbol log is more recent. It was created by John Napier, a Scottish mathematician, in the 1600’s. “Log” is short for logarithm, and represents the function that is the inverse of an exponential function. Use this idea to find the inverse for each of these functions. Write your answers in y = ___ form. a. y = log 9 (x) b. y = 10 x c. y = log 6 (x + 1) d. y = 52 x e. y = x5 f. y = 5 x +1 6. A LOGARITHM TABLE: Lynn was supposed to fill in this table for g(x) = log10 x , but her brother, Greg, had spilled orange juice on her calculator and the log button was stuck. She was fuming over how long it was going to take when her sister, Cheryl, suggested that she didn’t have to guess and check for all of them. She could get a few and then use what she knew about exponents to figure out the rest. a. With your team and without using the log button, use your calculator to estimate each value of g(x) to the nearest tenth. Lynn only had to use guess and check to find six of the results. See if your team can complete the table with fewer. b. For at least two of your entries where you did not use guess and check, explain your reasoning. c. What do you notice about the results for g(x) as x increases? x g(x) 0.01 0.1 0.5 1 0 2 3 4 5 6 7 8 9 10 1 20 30 100 2 d. Now that the table is complete, use the log button on your calculator to check your results. INVESTIGATING THE FAMILY OF LOGARITHMIC FUNCTIONS 7. Your Task: Investigate the family of logarithmic functions y = logb (x) . The questions below will help you investigate. Generate enough data that your team can find summary statements to present to the class. For each summary statement you find, prepare a transparency that shows and explains the summary statement. Include a justification for why your observation is true. How can you collect data for this family? How much data is enough? What have you learned about logarithms and inverses that can help you work with this family? How can “DrawInv” help? What patterns can you find in your team’s data? Why do they happen? What are all the domain for your function? What is the range? Are there some x values that do not make sense? Why or why not? Are all y-values possible? How can you be sure? What are some characteristics that all logarithmic functions have in common? What happens as the value of b changes? To begin your investigation of y = logb x each team member should choose a different positive value for b. Since there is no key for a logarithm of base b on your calculator, you will need to find another method to generate data for a table. a. Remember that a log is just the exponent that can be used with the base number to get x ! Make an x → y table for your base number. Use at least seven values for x . b. It may still be hard to make a table for your equation. Some x -values will be easier to work with than others. For example, if you chose y = log x and you want to find the 3 output when x = 10 , you will have to estimate to get about 2.1. How can you use your calculator to estimate? On the other hand if you use values like 1, 3, 9, 27 etc, or 1/3, 1/9, you can get an exact values for y . c. Once you have a table you can graph your equation. How can you check to see whether the graph is complete and correct? Complete you investigation of the family of functions of the form y = logb x . Use the discussion points above to guide your team’s discussion. USING LOGS TO SOLVE EXPONENTIAL EQUATIONS 8. Marta was convinced that there had to be some way to graph y = log 2 x on her graphing calculator. She typed in y = log(2 x ) and hit GRAPH “It WORKED!” Marta yelled in triumph. “Whaaaaat?” said Celeste. “I think y = log 2 x and y = log(2 x ) are totally different, and I bet we can prove it by converting both of them to exponential form.” “Yeah, I think you’re wrong, Marta,” said Sophia. “I think we can prove y = log 2 x and y = log(2 x ) are totally different by looking at the graphs.” a. Show that y = log 2 x and y = log(2 x ) are different by sketching the graph of y = log 2 x using what you learned in previous lessons. Then sketch what your grapher shows to be the graph of y = log(2 x ) . b. Now show that they are different by converting both of them to exponential form. 9. Sophia was intrigued. “The graph of y = log(2 x ) looks like a line. But is it really a line, or is the graphing calculator just fooling me again?” “Well, if it’s a line, it has to have a constant slope,” said Marta. “Why don’t we each pick two points and see if we all get the same slope?” a. Use Marta’s idea. To show that the graph of y = log(2 x ) is really a line, each team member should pick a different pair of points and use them to calculate the slope between those points. b. “Where did that number come from?” asked Sophia. “What does it have to do with log(2 x ) ? I wonder what the exact slope is.” Find a way to write the exact slope (that is, without decimals). c. Write an equation in y = mx + b form for your line. Use the exact version of the slope you found in part (b). 10. “The graph really helped me see that my idea doesn’t work,” said Marta. “I wonder if there is something special about y = log(2 x ) that made the graph come out to be a line, or whether this is also true for other numbers.” “That’s a good question,” said Celeste. “Let’s try graphing a few more.” a. For y = log(b x ) each team member should pick a different value of b that is a positive number and graph their equation on a graphing calculator. Sketch your results and label each graph with its equation. b. For each graph of y = log(b x ) , find the slope in its exact form (that is, without decimals). Use your slope to write an equation in y = mx + b form. 11. Generalize your results from the last problem. How can you transform the equations y = log(b x ) into an equivalent form using y = mx + b ? 12. THE RETURN OF AN IRRITATING PROBLEM Do you remember solving 1.04 x = 2 in your homework? What method(s) did you and your teammates use to find x? On tonight’s homework, there are several more of these problems. Don’t you wish there were a faster way so you could avoid the mess? 13. THERE MUST BE AN EASIER WAY It would certainly be helpful to have a method other than guess and check to solve equations like 1.04 x = 2 and you already know everything you need to develop it. a. What makes the equation 2 = 1.04 x so hard to solve for x? b. Surprise! You just discovered a method for getting rid of inconvenient exponents. Talk with your team about how your result from problem 5-94 can help you rewrite this equation. Be prepared to share your ideas with the class. c. Solve 2 = 1.04 x using this new method. Be sure to check your answer. 14. TRY IT OUT: Solve the following equations. Be accurate to three decimal places. Be sure to check your answers. a. 5 = 2.25 x b. 3.5 x = 10 c. 2(8 x ) = 128 d. 2x 8 = 128 e. What is different about part (e)? 15. CHANGING LOG BASES: Marta now knows that, if she wants to find log 2 (30) , she cannot just type log(2 30 ) into her calculator. She also knows that her calculator’s log key cannot do logs of base 2. Now how can she find what log 2 (30) equals? a. First, use your knowledge of logs to estimate log 2 (30) . b. Now look for a better estimate. Since you want to determine what log 2 (30) equals, you can write log 2 (30) = x . When working with a log equation, it is sometimes easier to convert it to exponential form. Convert this equation into its equivalent exponential form. c. Use the methods you developed earlier to solve this equation. Refer back to your work on problem 13. d. Congratulations! You have just evaluated a log base 2, even though your calculator does not know how to do that. First estimate an answer and then apply the method you have just developed to evaluate log 5 (200) . e. Describe how you would apply the process you used in part (d) to evaluate the expression log a b . INVESTIGATING OTHER LOG PROPERTIES 16. In problem 5-98, you created a table of the powers of 3 like the one at right. How can these values help you learn more about the properties of logarithms? If you do not have your table from problem 5-98, copy the table at right onto your paper. Then use this table to answer the questions below. Look for ways to show how you found your answer using the table. a. What is the value of (34 )(35 ) ? Use your calculator to find the answer. Is it necessary to multiply 34 by 35 to get the answer? Discuss ways that you could use both columns of the table at right to find the same answer without using a calculator (or doing any messy calculations by hand). Be sure to record your thinking on the table on your paper. b. x 0 1 2 3 4 5 6 7 8 9 3x 1 3 9 27 81 243 729 2187 6561 19683 b. See if your method from part (a) can help you find (33 )(35 ) . Be sure to verify your answer with your calculator. If your method does not work, use this additional solution to look for another way to use both columns of the table to multiply exponential expressions with the same base. c. Test your strategy two more times with problems you generate of the form (3a )(3b ) . Be ready to share your strategy with the class. 17. In problem 16, you developed a shortcut based on adding powers in an exponential table. Will a similar process work for adding logs? Since log 3 x is the inverse of 3x , it seems that you should be able to “inverse your logic” to simplify logarithmic expressions. Let’s try it. Find a strategy that is the “inverse” of the strategy you used in problem 5-110 and try it in part (a) below. a. Using the table of values, what is log 3 729 + log 3 27 ? What connection do you see between your answer and the product of 729 and 27? Verify your connection with the answer on your calculator. If your connection did not work, try another one. Keep going until you have found a shortcut to simplify the expression. Write down your method clearly. b. What about log 3 2187 + log 3 9 ? What is its value? Use the table to predict its value and find a way to justify your solution. How can log 3 2187 + log 3 9 be rewritten as a single logarithm? And how could you use the numbers 2187 and 9 to help you find that logarithm? c. Use your table strategy to find log 3 27 + log 3 243 and check your answer with your calculator. Now rewrite this expression as a single logarithm. How can 27 and 243 be used to rewrite this expression? d. What about subtracting logarithmic expressions? Find a way to rewrite log 3 6561 − log 3 729 so that it is a single logarithmic term that has the same value. Be ready to share your method with the class. 18. Problem 17 focused on logarithms in base 3. But what about logs of other bases, such as base 10? Write a conjecture on how the expressions log a + log b and log a − log b can each be rewritten as a single logarithmic expression. Then test your conjecture by rewriting the expressions below and testing your new expressions with a calculator. a. log 4 + log 5 b. log 18 − log 3 c. log 20 + log 5 d. log 13 − log 5 19. The fact that in any base m (when m > 0 ), log m a + log m b = log m ab is called that Product Property of Logarithms. How can you prove that this property is true? Follow the directions below to help you prove this property. a. Since logarithms are inverses of exponential functions, each of their properties can be derived from a similar property of exponents. You are trying to prove that “logs allow you to use sums to find products.” First, recall similar properties of exponents. If a = m x and b = m y , express a ⋅ b as a power of m . b. Rewrite a = m x , b = m y , and your answer to part (a) in logarithmic form. c. Substitute for x and y to obtain a log equation of base m that involves only the variables a and b . d. Similarly, the property log m a − log m b = log m ba is called the Quotient Property of Logarithms. Use a = m x and b = m y to express ba as a power of m. Then use a similar process to rewrite each into log form and prove the Quotient Property of Logs. 20. Use the properties of logs to write each of the following as a single logarithm, if possible. a. log1/2 (4) + log1/2 (2) − log1/2 (5) b. log 2 (M ) + log 3 (N ) c. log(k) + x log(m) d. 1 log 5 2 e. log(4) − log(3) + log(π ) + 3log(r) f. log(6) + 23 x + 2 log 5 (x + 1) 21. What values must x have so that the log(x) is a negative value. Justify your answer. USING EXPONENTIAL AND LOG FUNCTIONS: Applications 22. DUE DATE: Brad’s mother has just learned that she is pregnant! Brad wants to know when his new sibling will arrive and decides to do some research. On the Internet, he finds the following article: Hormone Levels for Pregnant Women When a woman becomes pregnant, the hormone HCG (human chlorionic gonadotropin) is produced in order to enable the baby to develop. During the first few weeks of pregnancy, the level of HCG hormone grows exponentially, starting with the day the embryo is implanted in the womb. However, the rate of growth varies with each pregnancy. Therefore, doctors cannot use just a single test to determine how long a woman has been pregnant. Commonly, the HCG levels are measured two days apart to look for this rate of growth. A woman who is not pregnant will often have an HCG level of between 0 and 5 mIU (milli-international units) per ml (milliliter). Then Brad remembered that his mother was tested for HCG during her last two doctor visits. On March 21, her HCG level was 200 mIU/ml, while two days later, her HCG level was 392 mIU/ml. a. Assuming that the model for HCG levels is of the form y = ab x , what equation models the growth of HCG for his mother’s pregnancy? b. Assuming that his mother’s the level of HCG on the day of implantation was 5 mIU/ml, how many days after implantation was his mother’s first doctor visit? What day did the baby most likely become implanted? c. Brad also learned that a baby is born approximately 37 weeks after implantation. When can Brad expect to become a big brother? 23. The context in problem 22 required you to assume that the exponential model had an asymptote at y = 0 to find the equation of the model. But what if the asymptote is not at the x-axis? Consider this situation below. a. Assume the graph of an exponential function passes through the points (3, 12.5) and (4, 11.25) . Is the exponential function increasing or decreasing? Justify your answer. b. If the horizontal asymptote for this function is the line y = 10 , make a sketch of its graph showing the horizontal asymptote. Verify that as x increases, the y values get closer to y = 10. c. If this function has the equation y = ab x + c , explain what the value of c would be, and then find an equation for your graph. 24. Janice would like to have $40,000 to help pay for college in 8 years. Currently, she has $1000. What interest rate, when compounded yearly, would help her reach her goal? a. What type of function would best model this situation? Explain how you know and write the general form of this function. b. If y represents the amount of money and x represents the number of years after today, find an equation that models her financial situation. What interest rate does she need to earn? c. Suppose she starts with $7800 and wants to have $18,400 twenty years from now. What interest rate does she need (compounded yearly) in this scenario? d. Which of the two scenarios described is more realistic? Justify your response. 25. THE CASE OF THE COOLING CORPSE: The coroner’s office is kept at a cool 17°C. Agent 008 kept pacing back and forth trying to keep warm as he waited for any new information about his latest case. For over three hours now, Dr. Dedman had been performing an autopsy on the Sideroad Slasher’s latest victim, and Agent 008 could see that the temperature of the room and the deafening silence were beginning to irritate even Dr. Dedman. The Slasher had been creating more work than Dr. Dedman cared to investigate. “Dr. Dedman, don’t you need to take a break?” Agent 008 queried. “You’ve been examining this dead body for hours! Even if there were any clues, you probably wouldn’t see them at this point.” “I don't know.” Dr. Dedman replied. “I just have this feeling something is not quite right. Somehow the Slasher slipped up with this one and left a clue. We just have to find it.” “Well, I have to check in with HQ,” 008 stated. “Do you mind if I step out for a couple of hours?” “No, that’s fine,” Dr. Dedman responded. “Maybe I’ll have something by the time you return.” Sure, 008 thought to himself. Someone always wants to be the hero and solve everything himself. The doctor just does not realize how big this case really is. The Slasher has left a trail of dead bodies through five states! 008 left, closing the door quietly. As he walked down the hall, he could hear the doctor’s voice fade away, as he described the victim’s gruesome appearance into the tape recorder. The hallway from the coroner’s office to the elevator was long and dark. This was the only way to Dr. Dedman’s office. Didn’t this frighten most people? Well, it didn’t seem to bother old Ajax Boraxo who was busy mopping the floor, thought 008 . . . most others wouldn’t notice, he reminded himself. He stopped to use the restroom and bumped into one of the deputy coroners. “Dedman still at it?” “Sure is, Dr. Quincy. He’s totally obsessed. He’s certain there is a clue.” As usual, leaving the courthouse, 008 had to sign out. “How’s it going down there, Agent 008?” Sergeant Foust asked. Foust spent most of his shifts monitoring the front door, forcing all visitors to sign in, while he recorded the time next to the signature. Agent 008 wondered if Foust longed for a more exciting aspect of law enforcement. He thought if he were doing Foust’s job he would get a little stir-crazy sitting behind a desk most of the day. Why would someone become a cop to do this? “Dr. Dedman is convinced he will find something soon. We’ll see!” Agent 008 responded. He noticed the time: Ten minutes before 2:00. Would he make it to HQ before the chief left? “Well, good luck!” Foust shouted as 008 headed out the door. Agent 008 sighed deeply when he returned to the courthouse. Foust gave his usual greeting: “Would the secret guest please sign in?” he would say, handing a pen to 008 as he walked through the door. Sign in again, he thought to himself. Annoying! 5:05 p.m. Agent 008 had not planned to be gone so long, but he had been caught up in what the staff at HQ had discovered about that calculator he had found. For a moment he saw a positive point to having anyone who came in or out of the courthouse sign in: he knew by quickly scanning the list that Dr. Dedman had not left. As he approached the coroner’s office, he had a strange feeling that something was wrong. He could not hear or see Dr. Dedman. When he opened the door, the sight inside stopped him in his tracks. Evidently, Dr. Dedman was now the newest victim of the Slasher. But wait! The other body, the one the doctor had been working on, was gone! Immediately, the security desk with its annoying sign-in sheet came to mind. Yes, there were lots of names on that list, but if he could determine the time of Dr. Dedman’s death, he might be able to scan the roster to find the murderer! Quickly, he grabbed the thermometer to measure the Doctor’s body temperature. He turned around and hit the security buzzer. The bells were deafening. He knew the building would be sealed off instantly and security would be there within seconds. “My God!” Foust cried as he rushed in. “What a travesty! I was talking with Dr. Dedman less than an hour ago.” As the security officers crowded into the room, Agent 008 explained what he knew. He had stopped long enough to check the doctor’s body temperature: 27°C. That was 10°C below normal. He figured that the doctor had been dead at least an hour. Then he remembered: the tape recorder! Dr. Dedman had been taping his observations; that was standard procedure. They began looking everywhere. The Slasher must have realized that the doctor had been taping and taken the tape recorder as well. Exactly an hour had passed during the search and Agent 008 noticed that the thermometer still remained in Dr. Dedman’s side. The thermometer clearly read 24°C. Agent 008 knew he could now determine the time of death precisely. When did Dr. Dedman die? Who is the murderer? Coroner’s Office - Please Sign In Name Time In Time Out Lt. Borman 12:08 2:47 Alice Bingham 12:22 1:38 Chuck Miranda 12:30 2:45 Harold Ford 12:51 1:25 Ajax Boraxo 1:00 2:30 D. C. Quincy 1:10 2:45 Agent 008 1:30 1:50 Ronda Ripley 1:43 2:10 Jeff Dangerfield 2:08 2:48 Stacy Simmons 2:14 2:51 Brock Ortiz 2:20 2:43 Pierce Bronson 3:48 4:18 Max Sharp 3:52 5:00 Maren Ezaki 3:57 4:45 Caroline Cress 4:08 4:23 Milly Osborne 4:17 4:39 D.C. Quincy 4:26 4:50 Vinney Gumbatz 4:35 Cory Delphene 4:48 4:57 Max Crutchfield 5:04 Agent 008 5:05 Security 5:12 Problem Solving, Reasoning, and Logarithms created by: Judith Kysh San Francisco State University [email protected] Materials adapted from CPM Algebra II Connections (2007) More information at www.cpm.org