Exponential and Logarithmic Functions

Transcription

1 C H A P T ER Exponential and Logarithmic Functions Scarlet macaws are native to the jungles of Southern Mexico and Central America, and can live up to 75 years. However, macaws and other birds are threatened by deforestation, which destroys their native habitat. You will use logarithmic functions to calculate the number of parrots in a shrinking rain forest..1 Logs, Exponents, and More Solving Exponential and Logarithmic Equations p Decibels, ph, and the Richter Scale Logarithms and Problem Solving Part I p Depreciation, Population Growth, and Radioactive Decay Logarithms and Problem Solving Part II p. 319 Money, Money, Money! Loans and Investments p. 327 Chapter l Exponential and Logarithmic Functions 307

2 30 Chapter l Exponential and Logarithmic Functions

3 .1 Logs, Exponents, and More Solving Exponential and Logarithmic Equations Objective In this lesson you will: l Solve exponential and logarithmic equations. Problem 1 Solving for the Result or Inverse Log Solve each logarithmic equation by first converting to an exponential equation. 1. log 2 x 5 2. log x 5 3. In x 5 4. log 3 x 3 5. log 5 x 0 6. log 6 x 2.34 Lesson.1 l Solving Exponential and Logarithmic Equations 309

4 7. log x ln x Problem 2 Solving for the Log or Exponent Solve each logarithmic equation by first converting to an exponential equation. 1. log 2 4 x 2. log 4 x 3. ln 4 x 4. log x 5. log x 310 Chapter l Exponential and Logarithmic Functions

5 Problem 3 Solving for Base Solve each logarithmic equation by converting to an exponential equation and applying the properties of exponents. 1. log x log x log x log x log x log x log x log x 9 2 Lesson.1 l Solving Exponential and Logarithmic Equations 311

6 Problem 4 More Solving Solve each logarithmic equation by converting to an exponential equation and applying the properties of exponents. 1. log 9,425 x 2. log 6 x ln x log x Be prepared to share your methods and solutions. 312 Chapter l Exponential and Logarithmic Functions

7 .2 Decibels, ph, and the Richter Scale Logarithms and Problem Solving Part I Objective In this lesson you will: l Use logarithms to solve problems. Key Terms l decibel l ph scale l Richter scale Problem 1 Decibels A decibel is a unit of measure for the loudness of sound. The formula for the loudness of a sound is given by db 10 log I I 0 where db is the decibel level. The quantity I 0 is the intensity of the threshold sound, a sound that can barely be perceived. The intensity of other sounds, I, are defined as the number of times more intense they are than threshold sound. 1. The sound in a quiet library is about 1000 times as intense as the threshold sound, I 1000I 0. What is the decibel level of a quiet library? Lesson.2 l Logarithms and Problem Solving Part I 313

8 2. The dial tone on a telephone has a decibel level of 0. A dial tone is how many times more intense than the threshold sound? 3. Prolonged exposure to sounds above 5 decibels can cause hearing damage or loss. A loud rock concert averages about 115 decibels. A rock concert is how many times more intense than the threshold sound? 4. A sound of a motorcycle is 100,000,000,000 times more intense than a threshold sound. What is the decibel level for a motorcycle? 314 Chapter l Exponential and Logarithmic Functions

9 Problem 2 ph The ph scale is a scale for measuring the acidity or alkalinity of a substance, which is determined by the concentration of hydrogen ions. The formula for ph is ph log H where H is the concentration of hydrogen ions. Solutions with a ph value of less than 7 are acidic. Solutions with a ph value greater than 7 are alkaline or basic. Solutions with a ph of 7 are neutral. For example, plain water has a ph of The H concentration in orange juice is What is the ph level of the orange juice? Is orange juice acidic or basic? 2. The concentration of hydrogen ions in baking soda is What is the ph level of baking soda? Is baking soda acidic or basic? 3. Vinegar has a ph of 2.2. What is the concentration of hydrogen ions in vinegar? Lesson.2 l Logarithms and Problem Solving Part I 315

10 4. Lime water has a ph of 12. What is the concentration of hydrogen ions in lime water? Problem 3 Richter Scale The Richter scale is a scale used to measure the intensity of earthquakes. A seismograph is an instrument that measures the motion of the ground. The formula for calculating the Richter scale is M log ( I I 0 ) where M is the magnitude of the earthquake on the Richter scale, usually rounded to the nearest tenth. The quantity I 0 represents the intensity of a zero-level earthquake the same distance from the epicenter. The seismographic reading for a zero-level earthquake is millimeter at a distance of 100 kilometers from the center of the earthquake. The intensity of other earthquakes, I, are defined as the number of times more intense they are than a zero-level earthquake. 1. The San Francisco earthquake of 1906 had a seismographic reading of 7943 millimeters registered 100 kilometers from the center. What was the magnitude of the San Francisco earthquake of 1906 on the Richter scale? 316 Chapter l Exponential and Logarithmic Functions

11 2. The San Francisco earthquake of October 17, 199 struck the Bay Area just before the third game of the World Series at Candlestick Park. It was the worst earthquake since Its seismographic reading of 12,59 millimeters was registered 100 kilometers from the center. What was the magnitude of the San Francisco earthquake of 199 on the Richter scale? 3. The great Alaska earthquake on March 27, 1940 had a magnitude of 9.2 on the Richter scale. What was its seismographic reading in millimeters 100 kilometers from the center? 4. Calculate the value of the seismographic reading for an earthquake of magnitude 7 on the Richter scale. Lesson.2 l Logarithms and Problem Solving Part I 317

12 5. Calculate the value of the seismographic reading for an earthquake of magnitude on the Richter scale. 6. How much greater is the motion between a magnitude earthquake and a magnitude 7 earthquake? What would you predict as the amount of damage caused by a magnitude compared to a magnitude 7 earthquake? Be prepared to share your methods and solutions. 31 Chapter l Exponential and Logarithmic Functions

13 .3 Depreciation, Population Growth, and Radioactive Decay Logarithms and Problem Solving Part II Objective In this lesson you will: l Use logarithms to solve real-world problems. Problem 1 Depreciation Some items, such as automobiles, are worth less over time. The age of an item can be predicted using the formula log t ( V C ) log(1 r ) where t is the age of the item in years, V is the value of the item after t years, C is the original value of the item, and r is the yearly rate of depreciation expressed as a decimal. 1. A sports car was originally purchased for $42,750 and is currently valued at $23,350. The average rate of depreciation for this car is 11.6% per year. How old is the car to the nearest tenth of a year? Lesson.3 l Logarithms and Problem Solving Part II 319

14 2. A compact car was originally purchased for $15,450 and is currently valued at $5250. The average rate of depreciation for this car is 25.5% per year. How old is this car to the nearest tenth of a year? 3. A 5-year-old car was originally purchased for $27,450. Its current value is $12,250. What is this car s annual rate of depreciation? 4. A 6-year-old car is currently valued at $10,250. The car depreciates at the rate of 27.5% per year. What was the original price of this car? 320 Chapter l Exponential and Logarithmic Functions

15 Problem 2 Population Growth 1. Some biologists study the population of a species in certain regions. The formula for the population of a species is n k log(a) where n represents the population of a species, A is the area of the region in which the species lives, and k is a constant that is determined by field studies. Based on population samples, a rainforest that is 1000 square miles has 2400 parrots. a. What is the value of k? b. Based on the current level of deforestation, it is estimated that in 10 years only about 200 square miles of the rainforest will remain. How many parrots will live in the rainforest in 10 years? Lesson.3 l Logarithms and Problem Solving Part II 321

16 2. Desalination is a process for producing fresh water from salt water. The amount of fresh water produced can be modeled using the formula y a b In t where t represents the time in hours, y represents the amount of fresh water produced in t hours, a represents the amount of fresh water produced in one hour, and b is the rate of production. a. In one desalination plant, 12.7 cubic yards of fresh water can be produced in one hour with a rate of production of How much fresh water can be produced after 5 hours? b. How long would it take for the plant to produce 100 cubic yards of fresh water? 322 Chapter l Exponential and Logarithmic Functions

17 3. The amount of medicine left in a patient s body can be predicted by the formula log t ( C A ) log(1 r ) where t is the time in hours since the medicine was administered, C is the current amount of the medicine left in the patient s body in milligrams, A is the original dose of the medicine in milligrams, and r is the rate at which the medicine leaves the body. a. A patient is given 10 milligrams of a medicine which leaves the body at the rate of 20% per hour. How long will it take for 2 milligrams of the medicine to remain in the patient s body? b. Six hours after administering a 20 milligram dose of medicine, 5 milligrams remain in the patient s body. At what rate is the medicine leaving the body? Lesson.3 l Logarithms and Problem Solving Part II 323

18 Problem 3 Radioactive Decay A radioactive isotope decays over time. The amount of a radioactive isotope remaining can be modeled using the formula A A 0 e kt where t represents the time in years, A represents the amount of the isotope remaining in grams after t years, A 0 represents the original amount of the isotope in grams, and k is a decay constant. The half-life,, of a radioactive isotope is the time required for half of the sample to decay. The relationship between k and is determined by the assumption that a sample of A 0 grams will contain 1 2 A grams after t years. 0 The half-lives of some common radioactive isotopes are shown in the table. Isotope Half-life, Decay Constant, k Decay Formula Uranium (U-23) 4,510,000,000 years Plutonium (Pu-239) 24,360 years Carbon (C-14) 5730 years 1. Determine an equation for the decay constant in terms of the half-life by substituting 1 2 A for A and solving for k Chapter l Exponential and Logarithmic Functions

19 2. Calculate the decay constant for each radioactive isotope in the table. Enter each decay constant in the table. 3. Write a decay formula for each radioactive isotope by substituting each decay constant into the general decay formula. Enter each decay formula in the table. 4. Calculate the percentage of Plutonium 239 remaining after 100,000 years. 5. Calculate the amount of 100 grams of Carbon 14 remaining after 10,000 years. Be prepared to share your methods and solutions. Lesson.3 l Logarithms and Problem Solving Part II 325

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21 .4 Money, Money, Money! Loans and Investments Objectives In this lesson you will: l Calculate the payment on a loan. l Calculate the total paid on a loan. l Calculate the number of payments. l Calculate the time it takes for an investment to reach a particular amount. Problem 1 Car Loans One of the first major purchases that a person makes is buying a car. Many people purchase cars by taking out a loan and making monthly payments until the loan is repaid with interest. The amount of the loan still owed is called the principal, the amount paid each month is the payment, and the interest rate is usually a yearly rate. Most car loans are for three to five years. You want to purchase a $16,500 car with a $1000 down payment. 1. What is the amount of the loan that you would need to purchase the car? The car dealership offers three different financing options. l A three-year loan at 3.6% l A four-year loan at 4.6% l A five-year loan at 5.6% The loan that is best for you depends on several considerations including: l How long do you want the loan to last? l What monthly payment can you afford? l How long are you planning on keeping the car? The monthly payment for a car loan can be determined using the formula ip p 0 1 (1 i ) n where i is the interest rate per payment period, P 0 is the original value of the loan, p is the monthly payment, and n is the number of payments. Lesson.4 l Loans and Investments 327

22 2. Calculate the monthly payment for each loan. Then calculate the total amount paid. a. A three-year loan at 3.6% b. A four-year loan at 4.6% c. A five-year loan at 5.6% 32 Chapter l Exponential and Logarithmic Functions

23 3. Based on the information in Question 2, which loan would you choose? Explain. 4. Your friend Herman heard that if you take a loan and pay more than the minimum payment you can save a lot of money. The number of monthly payments can be calculated using the formula log ( 1 ip 0 p ) n log(1 i ) where i is the interest rate per payment period, P 0 is the original value of the loan, p is the monthly payment, and n is the number of payments. a. How many payments would you make on the four-year loan if you paid $500 each month instead of the minimum payment? b. What is the total amount paid? Lesson.4 l Loans and Investments 329

24 c. How many payments of $500 would you make? What would be the amount of the last payment? d. How much money would you save by paying $500 each month instead of the minimum payment? 5. For each monthly payment amount on the five-year loan, calculate the number of monthly payments, the amount of the last payment, and the total amount paid. a. $300 b. $ Chapter l Exponential and Logarithmic Functions

25 Problem 2 Mortgages Besides a car, the largest purchase that many people make is a house, condominium, or apartment. The price of a home is usually much greater than the price of a car so nearly everyone takes out a mortgage or home loan of 60% to 95% of the selling price of the home. A standard mortgage usually requires a down payment of 20% of the selling price and is for a time period from 20 to 30 years. The median national price of a house in the year 2000 was $139, You want to purchase a house that is the median cost. What would be the amount of the down payment? What would be the amount of the loan? 2. You secure a 30-year mortgage loan at an annual percentage interest rate (APR) of 5.37%. a. What would be your monthly payment? b. What is the total amount that you would pay against the loan? c. How much interest would you pay over the length of the loan? Lesson.4 l Loans and Investments 331

26 3. You want your monthly payments on the 30-year loan to be $650. a. How many payments will you need to make? b. What is the total amount of the payments? c. How much is the amount of the last payment? d. How much interest would you save? 332 Chapter l Exponential and Logarithmic Functions

27 Problem 3 Investments You just got your first job. Your uncle Robert tells you that it s never too early to start planning for your retirement. He says you should start putting $10 into a savings account each month and in 40 years when you retire you will have thousands of dollars. You find a savings account that pays 3.2% monthly. The amount of money in the account can be calculated using the formula P p [(1 i ) i n 1] where P is the amount of money in the account, p is the amount that you deposit each month, n is the number of months, and i is the interest per payment period. 1. How much money would you have saved after 5 years? 2. How much money would you have saved after 40 years? 3. Was your uncle Robert correct? Would this be enough to retire? Explain. Lesson.4 l Loans and Investments 333

28 4. You deposit $10 each week instead of each month with the interest compounded weekly. How much money would you have saved after 40 years? You want to have $1,000,000 in 45 years when you retire. You want to know how many monthly deposits of $200 in this account will it take reach this goal. The formula for the number of months to reach an investment goal is log ( 1 if p ) n log(1 i ) where F is the future value or investment goal, p is the amount that you deposit each month, n is the number of months, and i is the interest per payment period. 5. How many payments of $200 a month would it take to reach your goal of $1,000,000? How many years is this? 334 Chapter l Exponential and Logarithmic Functions

29 6. How many payments of $500 a month would it take to reach your goal of $1,000,000? How many years is this? 7. What monthly payment would allow you to make your goal of $1,000,000 in 45 years? Be prepared to share your methods and solutions. Lesson.4 l Loans and Investments 335

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