Chapter 11 Logarithms Lesson 1: Introduction to Logs Lesson 2: Graphs of Logs Lesson 3: The Natural Log Lesson 4: Log Laws Lesson 5: Equations of Logs using Log Laws Lesson 6: Exponential Equations using Logs Lesson 7: Applications of Logs This assignment is a teacher-modified version of Algebra 2 Common Core Copyright (c) 2016 emath Instruction, LLC used by permission.
Lesson 1 Introduction to Logs Introduction to Logarithms: Exercise #1: Find the inverse of the function f(x) = 2 x. Recall: To find the inverse of a function, we and then solve for y. Logarithmic Functions: Defining Logarithmic Functions: A fancy word for exponent is logarithm (log). is the same as Log functions are the of exponential functions. Log Form: Exponential Form: 2 3 = 8 Notice the spiral effect: If a log function does not have a base, it is considered to be the common log. It has a base of 10.
Exercise #2: Rewrite the following in exponential form. (a) (b) (c) (d) Exercise 3: Rewrite the following in log form. (a) 4 2 = 16 (b) a 3 = 64 (c) c d = 7 (d) 8-1/2 = b
Exercise #4: Evaluate the following logarithms. If needed, write an equivalent exponential equation. (a) (b) (c) (d) (e) (f) (g) (h) (i) What do you notice about examples (g) (i)?
Exercise #5: Solving basic log equations. Solve each for x. If necessary round to the nearest hundredth. (a) (b) (c) (d) (e) (f) (g) (h) (i)
Lesson 1 Introduction to Logs HOMEWORK Fluency: 1.) What is the inverse of y = 6 x? (1) y = -6 x (3) x = log 6 y (2) y = log 6 x (4) x = y 6 2.) The value of log 5 167 is closest to which of the following? (1) 2.67 (3) 4.58 (2) 1.98 (4) 3.18 3.) Which expression could be used to determine the value of y in the equation? (1) (3) (2) (4) 4.) Rewrite the following logs in exponential form. (a) (b)
5.) Rewrite the following exponential functions in log form. (a) (b) 6.) Determine the value for each of the following logarithms. (a) (b) (c) (d) (e) (f) Solve for the unknown. When necessary round to the nearest hundredth. 7.). log 7 v = 4 8.) log r 1024 = 5 9.) log 5 625 = n
10.) log g 2 = (1/3) 11.) log 4 i = 5 12.) log 9 r = (3/2) 13.) log p 216 = 3 14.) log 6 q = -4 15.) log (1/2) (1/4) = x
Lesson 2 Graphs of Logs Exercise #1: Consider the logarithm function y = log 3 x and its inverse y = 3 x. (a) Construct a table of values for y = 3 x and then use this to construct a table of values for the function y = log 3 x. (b) Graph y = 3 x and y = log 3 x on the grid given. Label with equations. (c) State the natural domain and range of y = 3 x and y = log 3 x. y = 3 x y = log 3 x. Domain: Range: Domain: Range (d) Does this log function have an asymptote? Justify your answer.
Exercise 2: Using your calculator, sketch the graph of on the axes below. Label the x- intercepts. State the domain and range of. Domain: Range: Domain and Range of Basic Logarithmic Functions (without shifts):
Exercise 3: (a) Graph the function on the axes provided. (b) Graph the function shifted. on the same axes. Describe, in words, how the graph (c) Where is the asymptote in this graph?
Exercise 4: (a) Graph the function on the axes provided. (b) Graph the function shifted. on the same axes. Describe, in words, how the graph (c) Where is the asymptote in this graph?
Shifts in Log Functions: Horizontal Shifts: : shifts units : shifts units When there is a horizontal shift the asymptote will change. Vertical Shifts: : shifts units : shifts units Exercise 5: If the function following would represent its y-intercept? was graphed in the coordinate plane, which of the (1) 12 (2) 13 (3) 8 (4) 9 Exercise 6: The value of which functions x-intercept is larger f or h? Justify your answer.
Exercise 7: Which of the following equations describes the graph shown below? Show or explain how you made your choice. (1) y = log 3 (x + 2) 1 (2) y = log 2 (x 3) + 1 (3) y = log 2 (x + 3) 1 (4) y = log 3 (x + 3) 1 The fact that finding the logarithm of a non-positive number (negative or zero) is not possible in the real number system always use an inequality to find the domains of a variety of logarithmic functions. Exercise 8: Determine the domain of the functions below. State your answer in set-builder notation. (a) y = log 2 (3x 4) (b) y = log 3 (2x 5) (c) y = log(6 9x) (d) y = log (5x 10)
Lesson 2 Graphs of Logs HOMEWORK Complete the examples below. 1.) Which graph represents the function y = log 2 x? 2.) Which of the following represents the y-intercept of the function? (1) -8 (2) -5 (3) 3 (4) 5 3.) The domain of in the real numbers is (1) {x x > 0} (2) {x x > -5} (3) {x x > 5} (4) {x x > -4}
4.) Which of the following values of x is not in the domain of? (1) -3 (2) 0 (3) 5 (4) 4 5.) Which of the following is true about the function? (1) It has an x-intercept of 4 and a y-intercept of -1. (2) It has an 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 nd a y-intercept of -1. 6.) Which of the following represents a shift of 5 units right and 6 units up from the graph of f(x) = logx? (1) g(x) = log(x + 5) 6 (3) g(x) = log(x 5) 6 (2) g(x) = log(x + 5) + 6 (4) g(x) = log(x 5) + 6
7.) Determine the domains of each of the following logarithmic functions. State your answers using any accepted notation. (a) (b) (c) (d) 8.) The value of which functions x-intercept is larger f or h? Justify your answer.
9.) The graph of is shown below. (a) What is the value of b? Explain or show how you arrived at your answer. (b) The graph appears to pass through point (15,2) does it? Justify your answer. 10.) Graph the function on the graph provided. Be sure to label the asymptote. y x
Lesson 3 The Natural Log Because of the importance of y = e x, its inverse, known as the natural logarithm, is also important. The Natural Logarithm: The inverse of y = e x : y = lnx (y = log e x) The natural logarithm, like all logarithms, give an exponent as its output. In fact, it gives the power that we must raise e to in order to get the input. Exercise 1: Determine the values of each of the following. (a) (b) (c) (d) Exercise 2: Find the value of x in the examples below. If necessary, round to the nearest hundredth. (a) (b)
(c) (d) (e) (f) Exercise 3: A population of llamas on a tropical island can be modeled by the equation, where t represents the number of years since the llamas were first introduced to the island. (a) How many llamas were initially introduced at t = 0. Show calculations that lead to your answer. (b) Algebraically determine the number of years for the population to reach 600. Round your answer to the nearest tenth of a year.
Exercise 4: A hot liquid is cooling in a rook 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. T(m) = 101e -0.03m + 67 (a) What was the initial temperature of the water at m = 0? (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 o 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.
Lesson 3 The Natural Log HOMEWORK Answer all questions below. Be sure to show all work. 1.) Solve each equation for x. Round to the nearest hundredth. (a) e 0.52x = 18 (b) 53 + 4e -0.21x = 821 (c) 1.45 + 7e 0.32x = 34.23 (d) 6.3e -0.38x = 873.92 (e) 8lnx = 16 (f)2 ln 3x = 6
2.) Which of the following values of t solve the equation 5e 2t = 15? (1) (2) (3) (4) 3.) At which of the following values of x does have a zero? (1) (2) (3) (4) 4.) The savings in a bank account can be modeled using the equation, where t is the number of years the money has been in the account. Determine, to the nearest tenth of a year, how long it will take for the amount of savings to double from the initial amount deposited of $1250. 5.) An element has a decay constant of -0.0866 days. After how many days will 7.0 grams remain of a 10- gram sample? Use the exponential decay model A n = 10e -0.0866t, where A n is the amount remaining and t is time in days, to solve.
6.) Flu is spreading exponentially at a school. The number of new flu patients can be modeled using the equation, 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. Round to the nearest day. (b) Find the average rate of change of F over the first three weeks. Show the calculation that leads to your answer. Give proper units and round to the nearest tenth. What is the physical interpretation of your answer?
Lesson 4 Log Laws Logarithms have properties, just as exponents do. Since logarithms are essential exponents, logarithm laws are very similar to exponential laws. Expanding Logs: When we expand logs, we go from one log to many logs. Exercise 1: Expand the following logs. (a) (b) (c) (d)
(e) (f) Exercise 2: The expression is equivalent to (1) (3) (2) (4) Exercise 3: If 2x 3 = y, then logy equals (1) log(2x) + log3 (3) 3log2 + 3logx (2) 2log(2x) (4) log2 + 3logx Exercise 4: Which of the following is equal to log 3 (9x)? (1) log 3 2 + log 3 x (3) 2 + log 3 x (2) 2log 3 x (4) x + log 3 2
Exercise 5: The expression can be written in equivalent form as (1) 2logx 3 (3) 2logx 6 (2) log(2x) 3 (4) log(2x) 6 Exercise 6: Which of the following is equivalent to? (1) lnx + 6 (3) 3lnx 6 (2) 3lnx 2 (4) lnx 9 Condensing Logs When we expand logs, we go from many logs to one log. Exercise 7: Condense the following logs. (a) (b)
(c) (d) Exercise 8: If logx = 2loga + logb, then x equals (1) a 2 b (2) 2ab (3) a 2 + b (4) 2a + b Exercise 9: The expression is equivalent to (1) (2) (3) (4)
Exercise 10: If a = log3 and b = log2 then which of the following correctly expresses the value of log12 in terms of a and b? (1) a 2 + b (2) a + b 2 (3) 2a + b (4) a + 2b Exercise 11: If log2 = a and log3 = b, the expression is equivalent to (1) 2b a + 1 (2) 2b a 1 (3) b 2 a + 10 (4) Exercise 12: If log7 = k then log(4900) can be written in terms of k as (1) 2(k + 1) (2) 2k 1 (3) 2(k 3) (4) 2k + 1
Lesson 4 Log Laws HOMEWORK Answer the following questions. Show all work. 1.) Which of the following is not equivalent to log36? (1) (3) (2) (4) 2.) Which of the following is equivalent to? (1) (2) (3) (4) 3.) If log5 = p and log2 = q then log200 can be written in terms of p and q as (1) 4p + q (2) 2p + 3q (3) 2(p + q) (4) 3p + 2q 4.) The expression can be rewritten equivalently as (1) (2) (3) (4)
5.) If, then (1) 2k + 3 (2) 3k + 1 (3) k + 8 (4) k + 4 6.) Expand each of the log functions. (a) (b)
(c) (d) 7.) Condense each of the following log functions. (a) (b) (c) (d)
Using Rules of Logs to Solve Equations Lesson 5 Log Equations Using Laws of Logs Option 1: There are logs on BOTH sides of the equal sign 1. Condense both sides of the equation using the rules until each side is a SINGLE log expression. 2. Ignore the log and set the expressions equal to each other to solve. 3. Check - Log cannot be a zero or negative. Example:
Option 2: There is only ONE log in the equation 1. Condense the log side of the equation using the rules until it is a SINGLE log expression. 2. Rewrite the equation in exponential form and solve. 3. Check - Log cannot be a negative. Example: NOTE: THE CHECK IS REQUIRED!! Very often, log equations will reject one or more answers. Note that the can be anything, but the LOG cannot be zero or negative. Evaluate your check by plugging in your solutions.
Examples: Use the log rules and steps above to solve for x. 1. 2. 3. 4. 5. 6.
7. 8. 9. 10. 11. 12.
Lesson 5 Using Log Rules to Solve More Difficult Log Equations HOMEWORK Complete the following questions below. Show all work, including formulas. Solve for all values of x. 1.) 2.) 3.) 4.)
5.) 6.) 7.) 8.)
Solving Exponential Equations Using Logarithms: Lesson 6 Exponential Equations Using Logs Exercise 1: Solve each of the following equations for the value of x. Round your answers to the nearest hundredth. Example: Steps: (a) 5 x = 18 1.) Isolate the exponential function. 2.) Use log properties to rewrite. 3.) Isolate the x. 4.) Round to the indicated decimal place. (b) 4 x = 100 (c) 2 x = 1560
Exercise 2: Solve each of the following equations for the value of x. Round your answer to the nearest hundredth. (a) 6 x+3 = 50 (b) (1.03) x/2 5 = 2 (c) 4(2) x 3 = 17 (d) 17(5) x/3 = 4 Application of Logs (Log Word Problems): Many times, word problems involving logs use exponential functions. Recall the formulas: Growth: Decay:
Exercise 3: A biologist is modeling the population of bats on a tropical island. When he first starts observing them, there are 104 bats. The biologist believes that the bat population is growing at a rate of 3% per year. (a) Write an equation for the number of bats, B(t), as a function of the number of years, t, since the biologist started observing them. (b) Using your equation from part (a), algebraically determine the number of years it will take for the bat population to reach 200. Round your answer to the nearest year. Exercise 4: A stock has been declining in price at a steady pace of 5% per week. If the stock started at a price of $22.50 per share, determine algebraically the number of weeks it will take for the price to reach $10.00. Round your answer to the nearest week.
Exercise 5: A population of rabbits doubles every 30 days according to the formula:, where P is the population of rabbits on day t. How long will it take for the rabbit population to reach 360? Round to the nearest day. Exercise 6: The population of a small town is decreasing at the rate of 7.2% per year. The town historian records the population at the end of each year. In 2000 the population was 5,600. If this decrease continues, during what year will the population reach 3,300 people?
Lesson 6 Exponential Equations Using Logs HOMEWORK Answer all of the questions below. Show all work. 1.) Which of the following values, to the nearest hundredth, solves: 7 x = 500? (1) 3.19 (2) 3.83 (3) 2.74 (4) 2.17 2.) If there are initially 2200 bacteria in a culture, and the number of bacteria triple each hour, the number of bacteria after t hours can be found using the formula. How long will it take the culture to grow to 60,000 bacteria? (1) 5.06 hr (2) 4.25 hr (3) 3.01 hr (4) 1.52 hr 3.) To the nearest hundredth, the value of x that solves 5 x 4 = 275 is (1) 6.73 (2) 5.74 (3) 8.17 (4) 7.49
4.) Solve each of the following exponential equations. Round your answers to the nearest hundredth when necessary. (a) 9 x 3 = 250 (b) 50(2) x/2 = 1000 (c) 6 2x 5 = 300 (d) 5.) The population of Red Hook is growing at a rate of 3.5% per year. If its current population is 12,500 in how many years will the population exceed 20,000? Round your answer to the nearest year.
6.) A radioactive substance is decaying such that 2% of its mass is lost every year. Originally there were 50 kilograms of the substance present. (a) Write an equation for the amount, A, of the substance left after t-years. (b) Find the amount of time that it will take for half the initial amount to remain. Round your answer to the nearest tenth of year. 7.) Charlotte invests $500 into an account with 6.4% nominal yearly interest. How long will it take for the account to reach $1100. Round to the nearest tenth of a year.
Lesson 7 Applications of Logs Compound Interest: Recall: Compound interest is when the percent rate is applied more than once a year. This is known as compounding frequency. Formula: A(t) = P = r = n = Examples: Exercise 1: A person invests $350 in a bank account that promises a nominal rate of 2% compounded quarterly. (a) Write an equation for the amount this investment would be worth after t-years. (b) How much would the investment be worth after 20 years?
(c) Algebraically determine the time it will take for the investment to reach $400. Round to the nearest tenth of a year. (d) What is the effective annual rate for this investment? Round to the nearest hundredth of a percent. Exercise 2: Peter invests $5600 into an account with 3.4% nominal interest compounded monthly. (a) Write an equation for the amount this investment would be worth after t-years. (b) Algebraically determine the time it will take for the investment to triple. Round your answer to the nearest tenth of a year. (c) What is the effective annual rate for this investment? Round to the nearest hundredth of a percent.
Exercise 3: Christopher invests $6200 into an account with 4.3% nominal interest compounded daily. (a) Write an equation for the amount this investment would be worth after t-years. (b) Algebraically determine the time it will take for the investment to reach $10,000. Round your answer to the nearest tenth of a year. (c) What is the effective annual rate for this investment? Round to the nearest hundredth of a percent. Compounded Continuously: For an initial principal, P, compounded continuously at a nominal yearly rate of r, the investment would be worth an amount A given by: A(t) =
Exercise 4: A person invests $350 in a bank account that promises a nominal rate of 2% continuously compounded. (a) Write an equation for the amount this investment would be worth after t-years. (b) Algebraically determine the time it will take for the investment to reach $400. Round to the nearest tenth of a year. (c) What is the effective annual rate for this investment? Round to the nearest hundredth of a percent. Exercise 5: Louis invests $3050 in a bank account that promises a nominal rate of 2.6% continuously compounded. How long will it take for the investment to reach $6700? Round to the nearest tenth of a year.
Exercise 6: One of the medical uses of Iodine-131 (I-131), a radioactive isotope of iodine, is to enhance x-ray images. The half-life of I-131 is approximately 8.02 days. A patient is injected with 20 milligrams of I-131. Determine, to the nearest day, the amount of time needed before the amount of I-131 in the patient's body is approximately 7 milligrams. Exercise 7: The temperature T, of a given cup of coffee after it has been cooling for t minutes can be best modeled by the function below, where T 0 is the temperature of the room and k is a constant. (a) A cup of coffee is placed in a room that has a temperature of 70 o. After 4 minutes, the temperature of the coffee is now at 165 o. Compute the value of k to the nearest hundredth. Only an algebraic solution will be accepted. (b) Using this value of k, find the temperature, T of this cup of coffee that has been sitting in the room for a total of 12 minutes. Express your answer to the nearest degree. Only an algebraic solution will be accepted.
Answer all questions below. Show all work. Lesson 7 Applications of Logs HOMEWORK 1.) The amount of money in account with continuously compounded interest is given by the formula A=Pe rt, where P is principal, r is the annual interest rate, and t is the time in years. Calculate to the nearest hundredth of a year how long it takes for an amount of money to double if interest is compounded continuously at 3.1%. (1) 2.2 yr (2) 13.8 year (3) 22.4 yr (4) 15.5 yr 2.) Thomas invests $25,000 in an account that pays 4.75% annual interest compounded continuously. Using the formula A = Pe rt, where A = the amount in the account after t years, P = principal invested, and r = the annual interest rate, how many years, to the nearest tenth, will it take for Thomas s investment to triple? (1) 10.0 (2) 14.6 (3) 23.1 (4) 24.0 3.) An investment of $500 is made at 2.8% nominal interest compounded quarterly. (a) Write an equation that models the amount A the investment is worth t-years after the principal has been invested.
(b) Algebraically determine the number of years, to the nearest tenth of a year, that it will take for the investment to reach $800. 4.) After how many years, to the nearest tenth of a year, will $100 invested at an annual rate of 6% compounded continuously be worth at least $450? 5.) When Heather was 5 her grandparents gave her $5000 for her college education. Heather s parents invested that money into a college savings account earning 12% interest compounded monthly. How old will Heather be when her account reaches $20,000?
6.) Daniel is observing a mystery radioactive isotope. At 4 pm there are 2.5 grams and at 9 pm, there are 1.7 grams. What is the half-life? Round to the nearest hour.