Unit 2 Modeling with Exponential and Logarithmic Functions

Name: Period: Unit 2 Modeling with Exponential and Logarithmic Functions 1

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Investigation : Exponential Growth & Decay Materials Needed: Graphing Calculator (to serve as a random number generator) To use the calculator s random integer generator feature: 1. Type any number besides zero into your calculator, press STO, MATH,, ENTER, ENTER 2. Press MATH 5 1, 6 ) You can use numbers other than 1 and 6 here. The calculator will choose numbers between and including these numbers when you press enter. Continue pressing enter for more numbers. Investigation: 1. Choose a recorder to collect the class s data on the board. You ll copy the data down in your table later. 2. Everyone in the class should stand so that the recorder can count everyone and record the number of people standing in a table for Stage 0. 3. Use your calculator to find a random integer between 1 and 6. If you roll a 1, sit down. Before proceeding, allow time for the recorder to count the number of people still standing. When the recorder is finished counting, (s)he will let you know. 4. Repeat step 3 until fewer than 3 people are standing (or you run out of room on the table). 5. Record the data in your table. Stage 0 1 2 3 4 5 6 7 8 9 10 People Standing Questions: 1. What is your initial value for this set of data? What does it represent in the investigation? 2. Would it make more sense to find a common ratio (r) or common difference (d) for this data? Explain. 3. Based on your answer to Question 2, find the r OR d for the data you collected. Show the process you used to do so. 4. Could you estimate your answer to Question 3 without conducting the exploration? If so, how? 5. Write a recursive (NOW-NEXT) function that would help you make predictions for this data. 6. Write an explicit function using function notation that would help you make predictions for this data. In your function let x be the stage of the investigation and let f(x) equal the number of people standing in that stage. 3

Investigation 2: Half of a radioactive substance decays every 53 years. How much will remain of a 12 milligram sample after 530 years? Complete the table. Years 0 53 106 159 212 265 318 371 424 477 530 Remaining radioactive substance Questions: 7. What is your initial value for this set of data? What does it represent in the investigation? 8. Would it make more sense to find a common ratio (r) or common difference (d) for this data? Explain. 9. Based on your answer to Question 8, find the r OR d for the data you collected. Show the process you used to do so. 10. Could you estimate your answer to Question 9 without filling in the table? If so, how? 11. Write a recursive (NOW-NEXT) function that would help you make predictions for this data. 12. Write an explicit function using function notation that would help you make predictions for this data. In your function let x be the number of 53 year increments in the investigation and let f(x) equal the amount of radioactive substance remaining. 13. Write an explicit function using function notation that would help you make predictions for this data. In your function let x be the number of years in the investigation and let f(x) equal the amount of radioactive substance remaining. Use your equation to determine how much radioactive substance will remain after 500 years. 14. When will there be exactly 5 milligrams of the radioactive substance? Determine your answer to the nearest month. 15. Compare Investigation 1 and Investigation 2. What are the similarities and differences? 4

Investigation 3: You invest money in a savings account that earns 2.5% interest annually. How much money will you have at the end of 10 years if you begin with $1000? Complete the table. Years 0 1 2 3 4 5 6 7 8 9 10 Money in your account Questions: 16. What is your initial value for this set of data? What does it represent in the investigation? 17. Would it make more sense to find a common ratio (r) or common difference (d) for this data? Explain. 18. Based on your answer to Question 17, find the r OR d for the data you collected. Show the process you used to do so. 19. Could you estimate your answer to Question 18 without filling in the table? If so, how? 20. Write a recursive (NOW-NEXT) function that would help you make predictions for this data. 21. Write an explicit function using function notation that would help you make predictions for this data. In your function let x be the number of years in the investigation and let f(x) equal the amount of money in the account. 22. You find a bank that will pay you 3% interest annually, so you consider moving your account. Your current bank decides you re a good customer and offers you a special opportunity to compound your interest semiannually!!! (They make it sound like it s a really good deal, so you re curious). You don t play around with your money, so you ask what exactly that means. They explain that you ll still get 2.5% interest, but they ll give you 1.25% interest at the end of June and 1% interest at the end of December. You want to see if you make more money than you would if you just switched banks, so you do the calculations. Which bank is giving you a better deal? Explain your answer. 5

When you write an equation for a situation and use it to make predictions, you assume that other people who use it will understand the situation as well as you do. That s not always the case when you take away the context, so we sometimes need to provide some additional information to accompany the equation. 23. The domain of a function is the set of all the possible input values that can be used when evaluating it. If you remove your functions in Questions 6 and 13 and 21 from the context of this situation and simply look at the table and/or graph of the function, what numbers are part of the theoretical domain of the function? Will this be the case with all exponential functions? Why or why not? 24. When you consider the context, however, not all of the numbers in the theoretical domain really make sense. We call the numbers in the theoretical domain that make sense in our situation the practical domain. For instance, in the first investigation, our input values are Stages. If you look at the tables you created, what numbers would be a part of the practical domain for these investigations? Investigation 1 Investigation 2 Investigation 3 Make note of any similarities and differences and explain why they exist. 6

Summary of Findings The results of Investigation 1 & 2 simulate exponential decay because the values are being multiplied by the same value between 0 and 1 each time. The results of Investigation 3 simulate exponential growth because the values are being multiplied by the same value greater than 1 each time. As you may have noticed in Questions 4, the common ratio is related to the probability that you will be randomly assigned a 1 out of 6 options. Since this probability is 1:6, or 1 6, or 0.16, or 16. 6 % that fraction of the class will sit down. These numbers do not represent our common ratio, however. That s because when 1 of 6 people sit down, that leaves 5:6, or 5 or 0.83 or 83% of the class still standing. To find the common ratio 6 (b) using this probability (p) or any percentage for that matter, you can use the following equations. The probability or percent MUST be written as a decimal to use this formula. Exponential Growth b = 1 + p This is because you begin with 100% (1 when written as a decimal) and add the same percentage each time. Exponential Decay b = 1 p This is because you begin with 100% (1 when written as a decimal) and subtract the same percentage each time. In problems 5, 11, and 20, you wrote recursive functions for the exponential data. Make sure you include the starting value of your data (a) this will always be where the independent variable is 0. NEXT = NOW b, Starting at a y = ab x For each of these Investigations, we came up with functions with theoretical domains of all real numbers. In fact, all exponential functions will have domains of all real numbers. However, when we considered the context, we realized that the practical domain for these the first investigation only involves the whole numbers because all of the other real numbers would not work as stages. We also saw that in most scenarios where the domain is related to time, the domain is only positive values. 7

HW: ZOMBIES! SCENARIO 1 A rabid pack of zombies is growing exponentially! After an hour, the original zombie infected 5 people total, and those 5 zombies went on to infect 5 people, etc. After a zombie bite, it takes an hour to infect others. Develop a plan to determine how many newly infected zombies will be created after 4 hours. If possible, draw a diagram, create a table, a graph, and an equation. SCENARIO 2 During this attack, a pack of 4 zombies walked into town last night around midnight. Each zombie infected 3 people total, and those 3 zombies went on to infect 3 people, etc. After a zombie bite, it takes an hour to infect others. Develop a plan to determine the number of newly infected zombies at 6 am this morning. If possible, draw a diagram, create a table, a graph, and an equation. SCENARIO 3 At 9:00 am, the official count of the zombie infestation was 16384. Every hour the number of zombies quadruples. Around what time did the first zombie roll into town? 8

Guided Practice : Exponential Growth & Decay 1. You decide to conduct an investigation in a room of 100 standing students who were to randomly choose a number between and including 1 through 20. If they chose a multiple of 4, they were to sit down. You record the number of people still standing after each turn. a. What is the probability of choosing a multiple of 4? b. What is the common ratio for this investigation? c. What is the initial value of this investigation? d. Write a recursive equation for the investigation. e. Write an explicit equation for the investigation. f. Using your equations, how many stages of the investigation will occur before there are fewer than 3 people standing? 2. The amount of radioactive ore in a sample can be modeled by the equation y = 20(0.997) x, where x represents years and y is the amount of ore remaining in milligrams. a. What was the initial amount of radioactive ore? b. Is this an example of exponential growth or decay? c. What percentage of the ore is being lost or gained according to this model? d. When will there be half of the initial amount of the radioactive ore? 3. Complete the following table. Explicit Function Recursive Function Initial Value Common Ratio Growth or Decay? y = 2(3) x NEXT = NOW 0.5, Start at 3 0.125 4 y = ab x growth 9

4. The height of a plant can be modeled by the table below. Day 0 1 2 3 4 Height (in) 2.56 6.4 16 40 100 a) What is the initial height of the plant? Explain how you found your answer? b) What is the common ratio? Explain how you found your answer. c) Is this an example of exponential growth or decay? How could you find the answer if you only had the common ratio? d) Write the recursive function for this situation. e) Write the explicit function for this situation. f) How tall will the plant be on the 12 th day of data collection? How tall is this in inches, feet, yards, and miles? Does this seem realistic to you? g) On what day will the plant first be over 100 yards tall? 5. When you opened a savings account on your 15 th birthday, you deposited most of the money from your summer job ($2000). The banker who helped you informed you that you would received 1.5% interest each year. a. How much money will you have in the account when you turn 21? b. Use the properties of exponents to determine the monthly percentage interest rate that you could have gotten from the bank that would have given you the same amount of money when you turned 21. 6. For each scenario, write an explicit equation, define your variables, and determine the practical and theoretical domain. a) The town of Braeford was first established in 1854 when it had a population of 24. Since then it has grown by a percentage of 1.25% each year. b) A species of bacteria reproduces exactly once each hour on the hour. At this exact time, each organism present divides into two organisms. One of these bacteria is placed into a petri-dish at 8:00 this morning. 10

1. You buy a house for $130,000. It appreciates 6% per year. How much is it worth in 10 years? Question Exponential Growth or Decay? Write a function that represents this situation Initial Amount = Growth/Decay Rate: Percent = Decimal = Function Answer: 2. Justin Beiber is losing 20% of his hair each year. If he currently has 1,546 hairs on his head, about how many hairs will he have left after 10 years? Initial Amount = Growth/Decay Rate: Percent = Decimal = Function 3. If you invest $40 in an account for 10 years at a 3% interest rate compounded semi-annually, how much money will you have? Initial Amount = Growth/Decay Rate: Percent = Decimal = Function 4. A population of 100 frogs increases at an annual rate of 22%. How many frogs will there be in 5 years? Initial Amount = Growth/Decay Rate: Percent = Decimal = Function 11

Question 5. A species of extremely rare, deep water fish are slowly becoming extinct. If there are a total 821 of this type of fish and there are 15% fewer fish each month, how many will there be in half a year? Exponential Growth or Decay? Write a function that represents this situation Initial Amount = Growth/Decay Rate: Percent = Decimal = Function that represents this situation: Answer: 6. The population of Austin is growing at a rate of 5% per year. In 2010, the population was 500,000. What would be the predicted current population? Initial Amount = Growth/Decay Rate: Percent = Decimal = Function that represents this situation: 7. Use the equation from the previous question and predict in what year Austin s population will first reach 1,000,000. 8. Carbon-14 has a half-life of 5,730 years. If a fossil that originally had 500 mg of carbon-14 is found and determined to be 27,000 years old, how much carbon-14 was left? Initial Amount = Growth/Decay Rate: Percent = Decimal = Function that represents this situation: 9. A super-deadly strain of bacteria is causing the zombie population to double every 2 days. Currently, there are 25 zombies. After how many days will there be 25,600 zombies? Initial Amount = Growth/Decay Rate: Percent = Decimal = Function that represents this situation: (Careful with this one!!!) 12

HW : Exponential Growth & Decay 1. In 1990, Florida s population was about 13 million. Since1990, the state s population has grown about 1.7% each year. This means that Florida s population is growing exponentially. Year Population 1990 1991 1992 1993 1994 a) Write an explicit function in the form y = ab x that models the values in the table. b) What does x represent in your function? c) What is the a value in the equation and what does it represent in this context? d) What is the b value in the equation and what does it represent in this context? 2. Since 1985, the daily cost of patient care in community hospitals in the United States has increased about 8.1% per year. In 1985, such hospital costs were an average of $460 per day. a) Write an equation to model the cost of hospital care. Let x = the number of years after 1985. b) Find the approximate cost per day in 2012. c) When was the cost per day $1000 d) When was the cost per day $2000? 3. To treat some forms of cancer, doctors use Iodine-131 which has a half-life of 8 days. If a patient received 12 millicuries of Iodine-131, how much of the substance will remain in the patient 2 weeks later? 13

4. Suppose your parents deposited $1500 in an account paying 6.5% interest compounded annually when you were born. a) Find the account balance after 18 years. b) What would be the difference in the balance after 18 years if the interest rate in the original problem was 8% instead of 6.5%? c) What would be the difference in the balance if the interest was 6.5% and was compounded monthly instead of annually? 5. Since 1980, the number of gallons of whole milk each person in the US drinks in a year has decreased 4.1% each year. In1980, each person drank an average of 16.5 gallons of whole milk per year. Year Gal of milk 1980 a) Write a recursive function for the data in the table. 1981 1982 1983 b) Write an explicit function in the form y = ab x that models the values in 1984 the table. Define your variables. c) According to this same trend, how many gallons of milk did a person drink in a year in 1970? 6. The model y = 604000(0.982) x represents the population in Washington, D.C. x years after 1990. a) How many people were there in 1990? b) What percentage growth or decay does this model imply? c) Write a recursive function to represent the same model as the provided explicit function. d) Suppose the current trend continues, predict the number of people in DC in 2013. e) Suppose the current trend continues, when will the population of DC be approximately half what it was in 1990? 14

History of Logarithms (It s a great story) Quick Multiplication of large numbers Logarithmic tables Example: 2.93 x 1.26 From log table log 2.93 =.4669 use log table in reverse (antilog table or log -1 ) + log 1.26 =.1004 log -1.5673 = 3.69.5673 Your Turn: 2.13 x 2.81 15

Graphing Remember the chess board from Math 1? The following story has been told in different ways. Versions of the story date back over 1000 years. When the creator of the game of chess showed his invention to the ruler of the country, the ruler was so impressed that he gave the inventor the right to choose his own reward. The man asked the king for the following: that for the first square of the chess board, he would receive one grain of rice, two grains for the second square, four on the third square, and so forth, doubling the amount each time. The ruler quickly accepted the inventor's offer, and thought that the man hadn t asked for much. However, the king s treasurer explained that it would take more than all the rice in the kingdom to give the inventor the reward. The story usually ends with the inventor becoming the new king or being executed. Fill in the amount of rice in each square for the 1 st 40 squares 1 21 2 22 3 23 4 24 5 25 6 26 7 27 8 28 9 29 10 30 11 31 12 32 13 33 14 34 15 35 16 36 17 37 18 38 19 39 20 40 Graph as many grains of rice on the Cartesian plane (Vertical scale by 10 s) 16

Graph on semi-log paper 17

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The Richter Scale The magnitude of an earthquake is a measure of the amount of energy released at its source. The Richter scale is an exponential measure of earthquake magnitude, as shown on the right. The magnitude increases per unit as the energy released increases by powers of ten. A simpler way to examine the Richter Scale is shown below. An earthquake of magnitude 5 releases about 30 times as much energy as an earthquake of magnitude 4. Magnitude: +1 0 1 2 3 4 5 6 7 8 9 E E 30 1 E 30 2 E 30 3 E 30 4 E 30 5 E 30 6 E 30 7 E 30 8 E 30 9 Energy Released: X 30 In 1995, an earthquake in Mexico registered 8.0 on the Richter scale. In 2001, an earthquake of magnitude 6.8 shook Washington state. Let s compare the amounts of energy released in the two earthquakes. For the earthquake in Mexico at 8.0 on the Richter Scale, the energy released is E 30 8 and for the earthquake in Washington state, the energy released is E 30 6.8. A ratio of the two quakes and using the properties of exponents yields the following: Mexico Earthquake Washington Earthquale E 308 308 = = E 306.8 30 6.8 = 308 6.8 = 30 1.2 = 59.2 What this means is that the earthquake in Mexico released about 59 times as much energy as the earthquake in Washington. The exponents used by the Richter scale shown in the above example are called logarithms or logs. 19

Introduction to Exponential and Logarithmic Functions Notes Exponential Function: A function of the form y = a b x, where a 0, b > 0, and b 1. **Exponential Functions are functions whose equations contain a variable in the exponent!! Exponential Functions have the following characteristics: The functions is continuous and one-to-one The domain is the set of all real numbers The x-axis is an asymptote of the graph. The range is the set of all positive numbers if a > 0 and all negative numbers if a < 0. The graph contains the point (0, a). That is the y-intercept is a. The graphs of y = ab x and y = a ( 1 b )x are reflections across the y-axis. Examples: NOT Examples: Logarithmic Function: The function x = log b y, where b > 0 and b 1, is called a logarithmic function. This function is the inverse of the exponential function y = b x and has the following characteristics: The function is continuous and one-to-one. The domain is the set of all positive real numbers. The y-axis is an asymptote of the graph. The range is the set of all real numbers. The graph contains the point (1, 0). That is the x-intercept is 1. Logarithm: In general, the inverse of y = b x is x = b y. In x = b y, y is called the logarithm of x. It is usually written as y = log b x and is read y equals log base b of x. The inverse function of the exponential functions with base b, is called the logarithmic function with base b. For x > 0, b > 0, b 0, EXPONENTIAL FORM LOGARITHM FORM 20

Rewriting in both forms. Example 1) Rewrite logarithmic each equation in its equivalent exponential form. a. log 5 x = 2 d. 3 = log b 64 b. log 3 7 = y e. 3 = log 7 x c. 2 = log b 25 f. log 4 26 = y Example 2) Rewrite each exponential equation in its equivalent logarithmic form. a. 12 2 = x d. b 3 = 8 b. 2 5 = x e. b 3 = 27 c. 8 3 = c f. 4 y = 9 Basic and Inverse Log Properties- Because logs are exponents, they have properties that can be verified using the properties of exponents. Basic Properties: log b b = 1 because b 1 = b log b 1 = 0 because b 0 = 1 Inverse Properties: (Cancel with the same base!) log b b x = x b log b x = x Example 3) Evaluate using the log properties. a. log 7 7 e. log 9 9 b. log 5 1 f. log 8 1 c. log 4 4 5 g. 6 log 6 9 d. log 7 7 8 h. 3 log 3 17 21

Common Logarithm: Base 10 Logarithm, usually written without the subscript 10. log 10 x = log x, x > 0. Most calculators have a LOG key for evaluating common logarithms. The calculator is programmed in base 10. Example 4) Find the value of each log. Round to the nearest ten-thousandths. a. log 81,000 c. log 0.35 b. log 6 d. log 0.0027 Evaluating Logs using the Change of Base Formula For all positive numbers, a, b, and n, where a 1 and b 1, log a n = log b n log b a Example: log 5 12 = log 10 12 log 10 5 This formula allows us to evaluate a logarithmic expression of any base by translating the expression into one that involves common logarithms. Example 5) Evaluate each logarithm a. log 3 18 d. log 25 5 b. log 4 25 e. log 2 1024 c. log 2 16 f. log 5 125 Solving for variables with exponentials and logs. ****MAY HAVE TO REWRITE AND APPLY PROPERTIES OR CHANGE OF BASE FORMULA!!! Example 6) Solve for the variable: a. log 3 243 = y b. log 9 x = 3 c. log 8 n = 4 3 Example 7) Evaluate: a. log 8 8 4 = x b. log 9 9 2 = y Example 8) Solve each log equation. Be sure to check your answers! a. log 3 (3x 6) = log 3 (2x + 1) b. log 8 (x 2 14) = log 8 (5x) c. log 4 x 2 = log 4 (4x 3) d. log 5 (x 7) = 2 22

The Meaning of Logarithms (Kuta Software) 23

Graphing and Transforming Exponential Functions Part I Part 2: Horizontal Asymptote: an invisible line that the graph approaches as x goes to infinity or negative infinity. Identify the horizontal asymptote of (a): Identify the horizontal asymptote of (b): Identify the y-intercept of (a): End behavior: Identify the y-intercept of (b): End Behavior: 24

Transforming Exponential Functions Translate left or right: Vertical stretch or compression: Horizontal stretch or compression: Reflections: Translate up or down: Let s look at the following example. The graph on the right represents a transformation of the graph of f(x) = 3 log10 x + 1. x = 3: Stretches the graph vertically. h = 0: There is no horizontal shift. k = 1: The graph is translated 1 unit up. Graph the following function on the graph at right. Describe each transformation, give the domain and range, and identify any asymptotes. y = -2log10(x + 2) 4 Domain: Range: Asymptote: Description of transformations: 25

Independent Practice Graph the following transformations of the function y = log10 x on the coordinate planes. Determine the domain, range, and asymptotes of each transformation. Describe the transformations. 1) y = log10 x 6 6) y = -log10 (x + 2) 7) y = log10 2x Domain: Domain: Domain: Range: Range: Range: Asymptotes: Asymptotes: Asymptotes: Description: Description: Description: 2) Describe in your own words what happens to the graph of f(x) = log2 x under the given transformations, then graph. f(x) = log2 (x 2) b. f(x) = log2 (x) + 3 c. f(x) = log2 (x 2) + 3 3) State the domain, range, intercepts and asymptotes of f(x) = log2 (x 2) + 3. Domain: Range: Intercept(s): Asymptote(s): 4) Describe the transformations of y = 4 log2 (2x 4) + 6 from the parent function y = log2 (x). 5) Describe the transformations of y = -3 log 10 (4x + 3) 2 from the parent function y = log10 (x). 26

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Investigation 2: Solving for Exponents Logarithms can be used to find exponents that solve equations like 10 x = 9.5. For this reason, they are an invaluable tool in answering questions about exponential growth and decay. For example, the world population is currently about 6.2 billion and growing exponentially at a rate of about 1.14% per year. To find the time when this population is likely to double, you need to solve the equation 6.2(1.0114) t = 12.4, or (1.0114) t = 2 As you work on the problems of this investigation, look for ways to answer this question: How can common logarithms help in finding solutions of exponential equations? 1) Use number sense and what you already know about logarithms to solve these equations. a. 10 x = 1,000 b. 10 x + 2 = 1,000 c. 10 3x + 2 = 1,000 d. 2(10) x = 200 e. 3(10) x + 4 = 3,000 f. 10 2x = 50 g. 10 3x + 2 = 43 h. 12(10) 3x + 2 = 120 i. 3(10) x + 4 + 7 = 28 Use the rules of exponents or logarithms to find the value of x in each equation. 1. (5 x+1 ) 5 = 5 25x 2. (9 2x )(9 16 ) = 9 48 3. 450 440 = 4x 5 4. (64 2 )(16 x ) = 4 12 5. (16 1 2) (2 3 ) = x 6. (25 3 )(5 6 ) = 125 3x 2 7. 10 2x 1 = 100 8. -2(10) x+4 =.002 9. (10) x = 1 29

Use the rules of exponents or logarithms to find the value of x in each equation. 1. (2 12 ) 3 = 2 2x 2. (11 x )(11 15 ) = 11 105 3. 4x 434 = 420 4. (5 10 ) x = 5 50 5. (6 23 )(6 x ) = 6 57 6. 322 3x = 37 7. (6 2 )(36 x ) = 6 12 8. (4 1 2) (16 1 4) = x 9. (8 1 3) (64 3 2) = x 10. (25 3 )(5 9 ) = 125 x 11. (8 x )(2 5 ) = 4 7 12. (3 1 2) (27 1 2) = 9 x 13. 10 x = 10,000 14. 10 x+3 = 1,000 15. 10 3x+2 = 1,000 16. 10 2x = 50 17. 3(10) x+4 = 3,000 18. 12(10) 3x+2 = 120 30

Investigating the Properties of Logarithms **For the purpose of this activity we will be using Common Logs. PART I Complete the following table using your calculator. Round answers to four decimal places. log 5 0.6990 log 10 log 2 log 3 log 18 log 6 log 7 log 28 log 4 log ½ Log 3 log 6 log 2 Log 8 log 4 1. Using any patterns you see in the results above, what generalizations could be made? 2. How could we find the value of log 3 if the 3 button is missing from our calculator? PART II Complete the following table using your calculator. Round answers to four decimal places. log 12 log 6 + log 2 log 18 log 3 + log 6 log 9 log 3 + log 3 log 20 log 4 + log 5 log 26 log 2 + log 13 1. Using any patterns you see in the results above, what generalizations could be made? 2. Have you noticed a similar result before? Where? 3. How could we find the value of log 30 if the 3 button is missing from our calculator? 31

PART III Consider the following table. Number Equivalent value with a different base 16 2 4, 4 2 64 2 6, 4 3, 8 2 81 4 2 3, 9 49 2 7 25 2 5 Complete the following table using your calculator. Round values to four decimal places. 4log 2 1.2041 log 16 2log 5 log 25 3log 4 6log 2 log 64 2log 7 log 49 1. Using the above tables, what generalization(s) can be made? Can these generalizations be linked to your previous knowledge of exponents? How? 2. A student noticed that log ½ gave the same value as log 2. How is this possible? 3. Summarize all the properties of logarithms you know. Compare your results with others in the class. 32

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