LESSON 12.2 LOGS AND THEIR PROPERTIES

Transcription

1 LESSON. LOGS AND THEIR PROPERTIES LESSON. LOGS AND THEIR PROPERTIES 5

2 OVERVIEW Here's what ou'll learn in this lesson: The Logarithm Function a. Converting from eponents to logarithms and from logarithms to eponents b. Graphing a logarithmic function Logarithmic Properties Suppose ou see a newspaper ad for a savings and loan that promises to multipl our savings b 0 in 0 ears. Before ou open an account, ou want to know if the claim is true. Or, suppose a truck load of hazardous materials is accidentall spilled into a lake that supplies drinking water to our communit. Local health officials sa that the water will be safe to drink after 0 das, but ou want to be sure that this is true. In both of these situations, ou can use logarithms to help ou find the answer ou are looking for. In this lesson, ou'll learn about logarithms and how to graph logarithmic functions. In addition, ou'll learn some properties of logarithms. a. The algebra of logarithmic functions TOPIC THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS

3 EXPLAIN THE LOGARITHM FUNCTION Summar An Introduction to the Logarithmic Function You have alread worked with eponential functions, and ou have learned man of the properties of eponential functions. You have also learned how to graph eponential functions. For eample, Figure.. shows the graph of the eponential function =. Here is a table of ordered pairs that satisf = : Figure.. When ou graph =, ou begin b making a table of ordered pairs. To do this, ou choose values for, and then calculate their corresponding values for. For eample, if ou choose =, then = = 9. If ou go in the reverse direction that is, if ou start with a value for and calculate the corresponding value for then ou obtain a new function called a logarithm. The logarithmic function is the inverse function of the eponential function. That is, the logarithmic function "undoes" what the eponential function does. In the eample above, if ou choose = 9, then 9 =. To solve for, ou can write this as a logarithm: = log 9. Switching Between Eponential and Logarithmic Notation So ou now have two equivalent was of writing the same information. Here is a statement in eponential form: eponent = 9 base LESSON. LOGS AND THEIR PROPERTIES EXPLAIN 7

4 You can also write the same statement in logarithmic form: log 9 = logarithm base You've alread seen this idea of writing the same information in two different was when ou studied powers and roots. For eample, the statements = 9 and 9 = epress the same information. In both cases the base is. In eponential form the number is called the eponent. In logarithmic form it is called the logarithm. Here's a wa to generalize this: b L = is the same as = L These two statements are equivalent. The contain the same information written in different was. In both cases, b is the base and is a positive number and b. The is also a positive number. Here are some more eamples: Eponential Form b L = Logarithmic Form = L 5 = 5 log 5 5 = = log = 8 8 So an eponential statement can be written as a logarithmic statement. And an logarithmic statement can be written as an eponential statement. To do this:. Rewrite the statement, using the fact that b L = is the same as = L. For eample, to write = 8 in logarithmic form:. Rewrite the statement, using the fact that log 8 = b L = is the same as = L. Similarl, to write log 0 0,000 = in eponential form:. Rewrite the statement, using the fact that 0 = 0,000 b L = is the same as = L. Finding Logarithms B rewriting a logarithmic statement in eponential form, ou can find the value of the logarithm. Here are the steps:. Set the logarithm equal to L to create an equation. The variable L is the unknown value of the logarithm.. Rewrite the equation in eponential form.. Rewrite the constant term using the base.. Solve for L. 8 TOPIC THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS

5 For eample, to find log :. Set the logarithm equal to L to create an equation. log = L. Rewrite in eponential form. L =. Rewrite the constant term,, using the base,. L =. Solve for L. L = So, log =. 8 Here's a similar eample that uses fractions. To find log :. Set the logarithm equal to L to create an equation. log = L 8. Rewrite in eponential form. L =. Rewrite the constant term,, using the base,. L =. Solve for L. L = 8 So, log =. Graphing Logarithmic Functions You can graph a logarithmic function b using our knowledge of inverses and eponential functions. Here are the steps for graphing a logarithmic function:. Write the logarithmic function in the form =.. Switch and to get =. 8. Rewrite in eponential form, = b.. Graph this eponential function, = b. 5. Graph the original logarithmic function = b reflecting the eponential function = b about the line =. For eample, to graph the logarithmic function = log 5 :. The logarithmic function is in the form =. = log 5. Switch and. = log 5. Rewrite in eponential form, = b. = 5. Graph this eponential function, = 5. See Figure Graph = log 5 b reflecting = 5 about the line =. See Figure... 8 When the constant cannot be easil rewritten as a power of the base, ou will need to use a calculator. You'll learn how to do this in another lesson. Figure.. = Figure.. = 5 = log 5 LESSON. LOGS AND THEIR PROPERTIES EXPLAIN 9

6 Answers to Sample Problems Sample Problems. Write in logarithmic form: 7 = a. Rewrite the statement, using the fact that b L = log 7 = is the same as = L.. Write in eponential form: log 00 = a. = 00 b. L = 0 c. 5 d. 5 a. Rewrite the statement, using the fact that b L = is the same as = L.. Find: log 0 a. Set the logarithm equal to L to create an equation. b. Rewrite the equation in eponential form. c. Rewrite 0 using the base,. d. Solve for L.. Graph the function = log. a. Write the logarithmic function in the form = logb. b. Switch and. c. Rewrite in eponential form, = b. log 0 = L L = L = = log = log = d. d. Graph this eponential function, =. e. = = e. Graph = log b reflecting = about the line =. = log 50 TOPIC THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS

7 5. Graph the function = log. a. Write the logarithmic function in the form = logb. = log Answers to Sample Problems b. Switch and. c. Rewrite in eponential form, = b. = log c. = d. Graph this eponential function, =. d. e. Graph = log b reflecting = about the line =. e. = ( ) = = log LESSON. LOGS AND THEIR PROPERTIES EXPLAIN 5

8 LOGARITHMIC PROPERTIES Summar Logarithms can be rewritten using a variet of algebraic properties. These properties are like the properties that ou alread know for eponents. Here are the fundamental properties of eponents. Properties of Eponents Eamples b = b 7 =7 b 0 = 0 = b b = b = = b b 8 5 = b = 8 5 = b n = b n =8 5 7 = 8 5 Here are the corresponding fundamental properties of logarithms, as well as several others. Properties of Logarithms Eamples b = log 9 9 = = 0 log 7 = 0 Log of a Product: uv = u + v log 7 = log 7 + log Log of a Quotient: u v = u v log = log log 7 Log of a Power: u n = n u log 8 = log 8 b = 5 log 5 = v 7 = v log = log b n = n log 9 = 9 As with the properties of eponents, ou can use one or several properties of logarithms to rewrite statements that contain logarithms. Here are some eamples. To find log 7 7 :. Use the propert b n = n. log 7 7 = So, log 7 7 =. 5 TOPIC THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS

9 To rewrite the epression log [ (5 + 7)]:. Use the log of a product propert. log [ (5 + 7)] = log + log (5 + 7). Use the log of a power propert. = log + log (5 + 7) So, log [ (5 + 7)] = log + log (5 + 7). To rewrite z as a single logarithm:. Use the log of a power propert z = 5 + z 7. Use the log of a quotient propert. = + z 7 5 You have to work from left to right when ou rewrite logarithms. In this eample, ou subtract the two logs, then ou add.. Use the log of a product propert. = z 7 5 So, z =. z 7 5 Sample Problems Answers to Sample Problems. Simplif: log 7 ( 7) a. Use the propert logb b =. log 7 ( 7) =. Simplif: log (z + 9) a. Use the propert b log b =. log (z + 9). Rewrite this epression: log 7 p qr = z + 9 a. Use the log of a quotient propert. log7 p qr b. Use the log of a product propert.. Write log 9 using log 9. = log 7 p log 7 qr = b. log 7 + log 7 p (log 7 q + log 7 r) or log 7 + log 7 p log 7 q log 7 r a. Use the propert = v. log 9 v = a. log 9 5. Write as a single logarithm not containing eponents: log w f 7 + log w g 7 a. Use the log of a product propert. log w f 7 + log w g 7 b. Use the log of a power propert. = _ = a. log w f 7 g 7 or log w (fg) 7 b. 7log w (fg) LESSON. LOGS AND THEIR PROPERTIES EXPLAIN 5

10 Answers to Sample Problems Sample Problems EXPLORE On the computer ou used the grapher to graph various logarithmic functions and compared their behavior. You also eplored the algebraic properties of logarithms. Below are some additional problems to eplore.. Graph these logarithmic functions on the same set of aes, and answer the questions below. = log = log = log e a. To graph the functions, start Function Inverse b finding the inverse function = log = of each logarithmic function. = log = = log e = e b. Then graph each eponential function. = = e = c. = = log = log e = log c. Finall, reflect the graph of each eponential function about the line = to graph each corresponding logarithmic function. = = e = d. > 0 Notice b looking at the graphs that the are not defined for 0. d. What is the domain of each of the logarithmic functions? 5 TOPIC THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS

11 e. What is the range of each of the logarithmic functions? f. Find the point which is common to all three logarithmic graphs. g. For >, list the three logarithmic graphs in order of their distance from the -ais. Start with the closest one. h. For <, list the three logarithmic graphs in order of their distance from the -ais. Answers to Sample Problems e. All real numbers. Notice b looking at the graphs that -values can be positive or negative. f. (, 0) g. = log = log e = log h. = log = log e = log. Graph these logarithmic functions on the same set of aes, and answer the questions below. = log = log a. To graph the functions, start Function Inverse b finding the inverse function = log = of each logarithmic function. = log = b. Then graph each eponential function. = = c. Finall, reflect the graph of each eponential function about the line = to graph each corresponding logarithmic function. = = c. = = log = log d. What is the domain of each of the logarithmic functions? d. > 0 Notice b looking at the graphs that the are not defined for 0. LESSON. LOGS AND THEIR PROPERTIES EXPLORE 55

12 Answers to Sample Problems e. All real numbers. Notice b looking at the graphs that values can be positive or negative. f. (, 0) e. What is the range of each of the logarithmic functions? f. Find the point which is common to both logarithmic graphs. g. Increase. Decrease. g. Does the graph of = log increase or decrease as ou move from left to right along the -ais? How about = log? M 000. This is a formula for political fund raising: C = a + b log 0. Here, the variable M describes the number of funding requests that have been mailed, and the variable C describes the number of contributions received in response to the requests. The numbers a and b are constants which are determined b the nature of the request and the contents of each specific mailing. Here is a graph which illustrates the formula for a given a and b. C 0, ,000 00,000 M a. Use the graph to find the number of When M = 000, contributions, C, corresponding C = to M = 000 mailings. b. 0,000 b. Use the graph to find the number of When M =, mailings, M, required to produce C = 000. C = 000 contributions. c. The contributions increase rapidl at first, but eventuall taper off. c. What happens to the contributions as as the number of mailings increase? 5 TOPIC THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS

13 . Calculate the following logarithms: log, log 8, log, log. Use our answers to illustrate the log of a product propert, the log of a quotient propert and the log of a power propert, respectivel. Answers to Sample Problems a. Calculate log. Set log equal to. log = Rewrite the epression in eponential form. = Solve for. = b. Calculate log 8. Set log 8 equal to. Rewrite the epression in eponential form. Solve for. c. Calculate log. Set log equal to. Rewrite the epression in eponential form. Solve for. d. Calculate log. Set log equal to. Rewrite the epression in eponential form. Solve for. = = = b. log 8 = = 8 c. log = = d. log = = 5 e. Illustrate the log of a log + log 8 = log ( 8) product propert. Substitute = log the values ou found in (a), + = 5 (b), and (d) for log, log 8, and log. f. Illustrate the log of a quotient log log 8 = log 8 propert. Substitute the values = log ou found in (d), (b) and (a) for log, log 8, and log. = g. Illustrate the log of a power log = log propert. Substitute the values = log ou found in (a) and (c) for log and log. = f. 5,, g., LESSON. LOGS AND THEIR PROPERTIES EXPLORE 57

14 HOMEWORK Homework Problems Circle the homework problems assigned to ou b the computer, then complete them below. Eplain The Logarithm Function. Write this eponential statement in logarithmic form: 0 = Find: log 5 5. In order to graph the function = log 7, ou can use its inverse function. What is this inverse function?. Write this logarithmic statement in eponential form: log 500 = 5. Find: log. Graph the function = log. 7. Write this eponential statement in logarithmic form: 7 = 8. Find: log.. 9. Suppose that when ou graduate from college, ou deposit one dollar in a savings account for our retirement, t ears later. The final amount in our savings account after t ears is given b A. If ou receive 7% interest compounded annuall, the formula which tells ou how man ears, t it takes to accumulate the amount A is t = log.07 A. Graph this function. 0. The formula for the rate of deca of a radioactive chemical is R given b T = log B. Here, S is the starting amount of the S chemical in grams and R is the amount of the chemical in grams remaining after T ears. Write this formula in eponential form. Logarithmic Properties. Find: log 7 7. Rewrite as a single logarithm: log d + log d 5. Rewrite using the log of a power propert: log. Simplif: 0 log 0 abc 7. Rewrite using the log of a quotient propert: log 8. Rewrite using the log of a product propert and the log of a power propert: log 5 9. Simplif: log 0. Rewrite using the log of a product propert to get an epression with four terms: log B 7z. The magnitude of an earthquake of intensit I as compared to one of minimum intensit M is measured on the Richter I ab 7cd scale as R = log 0. Use a propert of logarithms to M rewrite this formula for R in terms of logarithms that do not contain fractions.. The ph of a particular fruit juice is given b the formula = log Find to four decimal places, given that log 0.5 = 0.9. (Hint: Use properties of logarithms to get our answer.). Find: log. Write as a single logarithm: log u + 5log v 8log w. Find: log. Graph the function = log. 58 TOPIC THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS

15 Eplore 5. The following functions are graphed on the grid in Figure..: = log = log 5 = log 7 Label each graph with the appropriate function. Figure... The following functions are graphed on the grid in Figure..8: = log 5 = log 5 Label each graph with the appropriate function. About what line can ou reflect one graph in order to get the other? a b c 7. The log of a product propert allows ou to rewrite log as log + log. Use numbers which are powers of to: a. rewrite log in a different wa using the Log of a Product Propert. b. rewrite log using the Log of a Quotient Propert. c. rewrite log using the Log of a Power Propert. 8. Describe how the graph of the logarithm = changes as ou increase the value of the base b for b >. Use the graph of = log 5 shown in Figure.. to help ou. Figure.. = log 5 9. Compare a logarithmic graph = with base b < and another logarithmic graph with base b >. What is the same on both graphs? 0. Circle the epressions below that are equal. log log log a b Figure..5 LESSON. LOGS AND THEIR PROPERTIES HOMEWORK 59

16 APPLY Practice Problems Here are some additional practice problems for ou to tr. The Logarithm Function. Write this eponential statement in logarithmic form: = 8. Write this eponential statement in logarithmic form: 5 = 0. Write this eponential statement in logarithmic form: 5 = 5. Write this eponential statement in logarithmic form: = 5 5. Write this eponential statement in logarithmic form: = a. Write this eponential statement in logarithmic form: b = 7. Find: log 8. Find: log 7 9. Find: log 8 0. Find: log 7 9. Find: log 5. Find: log. What is the inverse of the function f ( ) = log?. What is the inverse of the function f ( ) = log 7? 5. What is the inverse of the function f ( ) = log?. What is the inverse of the function f ( ) = log.5? 7. What is the inverse of the function f ( ) =? 8. What is the inverse of the function f ( ) = log a? 9. Write this logarithmic statement in eponential form: log 5 5 = 0. Write this logarithmic statement in eponential form: 5 log =. Write this logarithmic statement in eponential form: log 8 5 =. Write this logarithmic statement in eponential form: 8 =. Write this logarithmic statement in eponential form: log = 5. Write this logarithmic statement in eponential form: log 0 = 5. Graph the function = log.. Graph the function = log. 7. Graph the function = log. 8. Graph the function = log. Logarithmic Properties 9. Rewrite using the log of a power propert: log 0. Rewrite using the log of a power propert: a 8. Rewrite using the log of a power propert: 7log 5. Rewrite using the log of a quotient propert: log 7. Rewrite using the log of a quotient propert:. Rewrite using the log of a quotient propert: log 5 5. Rewrite using the log of a product propert to get an epression with two terms: log 5. Rewrite using the log of a product propert to get an epression with three terms: 5mn 7. Rewrite using the log of a product propert to get an epression with three terms: 8. Rewrite as a single logarithm: log c 5 + log c 7 9. Rewrite as a single logarithm: + 5 m 7 8 z 0 TOPIC THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS

17 0. Rewrite as a single logarithm: + z. Rewrite as a single logarithm: log 5 7 log 5. Rewrite as a single logarithm: log 9 5 log. Rewrite as a single logarithm: log 7 log 7. Rewrite as a single logarithm: log a + 7log b log c 5. Rewrite as a single logarithm: log a + 5log b log c. Rewrite as a single logarithm: a log b log c log z 7. Simplif: log 8. Simplif: 7 log 7 9. Simplif: log ( ) 50. Simplif: 5 log 5 5. Simplif: log z (z ) 5. Simplif: (m ) log m 5. Rewrite using the log of a quotient propert and the log of a z power propert: log 5 5. Rewrite using the log of a product propert and the log of a power propert: log Rewrite using the properties of logarithms to get an epression with three terms: log 5. Rewrite using the properties of logarithms to get an epression with three terms: log z LESSON. LOGS AND THEIR PROPERTIES APPLY

18 PRACTICE TEST EVALUATE Take this practice test to make sure ou are read for the final quiz in Evaluate.. Write this eponential statement in logarithmic form: 8 = z. Find the value of log.. Graph the function = log 5.. The formula for the growth rate of an eperimental bacterium is given b the eponential function P = Ie kt. Here, I is the starting population number, P is the population after t ears, and k is a proportionalit constant determined b the laborator conditions. Write this formula in logarithmic form. 5. Simplif: 5 log 5 z. Using the log of a product propert twice, rewrite: log 9 9AB 7. Write as a single logarithm: log w log w log w 8. Use properties of logarithms to rewrite as an epression with three terms: Graph the functions = log a and =, where ou know that a and b are greater than and a > b. Use the grid in Figure Consider the graphs shown in Figure..8. a. Identif which graph represents an eponential function with base less than. b. Identif which graph represents a logarithmic function with base greater than. c a b d Figure..8. Use properties of logarithms to determine whether the following is true: log 5 5 = log 5 5. Suppose that ou have a formula to measure the loudness of sounds on a logarithmic scale. You do not know what base to use for our logarithms, but ou have determined b eperiment that =. and that = 5.. Use this information and properties of logarithms to calculate. (, 0) Figure..7 TOPIC THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS