NCTM Content Standard/National Science Education Standard:

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1 Tile: Logarihms Brief Overview: In his Concep Developmen Uni, he concep of logarihms is discussed. The relaionship beween eponenial equaions and logarihmic equaions is eplored. The properies of logs are discussed and applied. Sudens will pracice he new skills using cooperaive learning games and real world applicaions. NCTM Conen Sandard/Naional Science Educaion Sandard: undersand relaions and funcions and selec, conver fleibly among, and use various represenaions for hem undersand and compare he properies of classes of funcions, including eponenial, polynomial, raional, logarihmic, and periodic funcions Grade/Level: Grades 9 /Algebra, College Algebra, Pre Calculus Duraion/Lengh: Three 90-minue class periods Suden Oucomes: Sudens will: Rewrie eponenial epressions in logarihmic form and vice versa Apply properies of logarihms Solve eponenial and logarihmic equaions Maerials and Resources: Inde cards Graphing calculaors Workshees o Concenraion Game wih Logarihms o Basic Logarihms Flash Cards o Advanced Logarihms Card Game o Reeaching Properies of Logarihms o Laws of Logarihms o Basic Eponenial and Logarihmic Equaions Round Table Game o Laws of Logarihms in Logarihmic Equaions Coach/Player Game o Solving Eponenial and Logarihmic Equaions o Applicaions of Logarihms Logarihms

2 Developmen/Procedures: Lesson Preassessmen Sudens are epeced o know how o graph funcions and read values off of he graph. They should also be able o solve polynomial equaions and be familiar wih he properies of eponens and radicals. They have already had a lesson on e. Disribue graphing calculaors. Assess he sudens abiliy o apply hese skills by posing he following quesion: Five sudens cach he flu a he end of winer break. They come o school conagious afer New Year s Day. The spread of he virus can be modeled by N 5+.5, where N is he number of people infeced afer days. a) Graph he funcion using your calculaor. b) Find how many sudens will be infeced afer days? Answer: 0 c) The school will close if 00 sudens are sick. How many days afer he winer break will his occur? Answer: 7 Launch In he previous eample, daa values can be easily read off of he graph or able of he funcion. Ask sudens o algebraically check he answer o par (b). Now invie sudens o solve par (c) algebraically. Sudens will run ino he difficuly of how o solve for he. Ask sudens wha sraegy hey hink migh help hem 95.5 solve for he. Possible suden responses a his poin migh be dividing by.5 or aking an h roo. Emphasize ha wha he sudens are rying o figure ou is o wha power hey have o raise.5 in order o ge 95. Teacher Faciliaion Look a a simpler eample ha could be solved by guess and check. Wha does have o equal in he problem 9? Sudens should quickly be able o say ha. Logarihms

3 Inroduce he idea of rewriing he eponenial epression in logarihmic form. To solve 9 for, rewrie i in logarihmic form: log 9. Pracice rewriing several eponenial epressions as logarihms and vice versa wih he enire class: 5 log 5 log6 6 log log m log 8 m 8 Ans: 8 y log 8 y Ans: log w w Ans: 6 a 6 log 6 6 a Ans: ½ Suden Applicaion Have he sudens pracice wriing eponenial equaions ino logarihmic equaions by playing he Concenraion Game wih Logarihms. Embedded Assessmen Observe suden inpu during pracice problems o assess wheher hey undersand he concep. Sudens will be helping each oher o ge he correc maches during he game. During he card game, he eacher circulaes hrough he room and assiss sudens if hey need help. Sudens can be asked o wrie down he pairs hey creaed and urn in he paper o he eacher afer he game o assess heir undersanding. Reeaching/Eension Use Basic Logarihm Flashcards ha gradually increases in difficuly for sudens who need reeaching. Use he Advanced Logarihms Card Game for sudens who undersand he concep. Sudens who undersand he concep can also work ogeher o make a new se of game cards. Logarihms

4 Lesson Preassessmen Use 5 of he concenraion game cards from he previous lesson (some eponenial and some logarihmic) and have sudens wrie he opposie form. Launch Have sudens eamine he buons on a scienific or graphing calculaor o deermine which logs can be done on he calculaor. Ask sudens o ry o evaluae he following logs o ge he correc answer on he calculaor wih hose logarihm buons: ln5.708 log0 000 log55 Sudens should be able o complee he firs eamples, bu will ask how o ener he bases since he calculaor only has a log buon. They will no be able o ge he calculaor o evaluae he las eample. Teacher Faciliaion Discuss ha he calculaor can only evaluae log base e (which is also called naural log and uses he noaion ln) and log base 0 (which is called common log and uses he noaion log). Discuss he change of base formula, which allows you o change any base ino he wo bases he calculaor can evaluae: log b ln b loga b log a ln a Now sudens will be able o change he base of he hird eample from he Launch and evaluae i: log5 ln5 log55 log 5 ln 5 Insruc he sudens o evaluae he numeraor and denominaor separaely and hen divide. Have hem ry boh he common log and he naural log, so ha hey see ha he inermediae resuls will look differenly, bu he final answer will be he same. This will help hem undersand ha hey can choose any base. Discuss he oher hree properies of logarihms and relae hem o he properies of eponens ha sudens should already know. Consider placing he properies on senence Logarihms

5 srips or poser paper o hang around he classroom, leaving hem up for fuure reference. Produc Propery: Concep: log ab can be rewrien as log a+ logb a b a b Compare o: + Quoien Propery: Concep: log c can be rewrien as log d c log d Compare o: a b a b Power Propery: Concep: log m n can be rewrien as a b a b Compare o: ( ) nlog m Pracice rewriing several logarihmic epressions using he properies (boh epanding and collapsing): log 5 log 5 Change of base: log log 5y Produc Propery: log 5 + log + log y ln Quoien Propery: ln ln 7 7 log w Power Propery: log w log Produc Propery and Change of log log Base: log + log + log log ln + ln y ln + ln y Produc Propery and Power Propery: ln y 7 wz log Quoien, Produc, and Power Properies: log wz log 5 log w + log z log log w+ 7 log z log Suden Applicaion All sudens will paricipae in he Concenric Circles aciviy o pracice he properies. Embedded Assessmen Observe suden inpu during pracice problems. Sudens will be helping each oher o ge he correc answers during he Concenric Circle game. Logarihms 5

6 During he card game, sand in he middle of he circles and assis where necessary. Reeaching/Eension Use Reeaching Properies of Logarihms ha gradually increases in difficuly for sudens who need reeaching. Use he Laws of Logarihms for sudens who undersand he concep. Sudens who undersand he concep can also work ogeher o make a new se of game cards. Lesson Preassessmen Use 5 of he concenric circle game cards from he previous lesson and insruc sudens o wrie eiher he epanded or collapsed form of he epression. Launch Reurn o he flu eample from Lesson. Ask sudens o solve par (b) by rewriing i in logarihmic form and using he properies of logs log.5 95 log 95 log Sudens have jus solved an eponenial equaion by using he skills hey have learned over he pas wo lessons. Teacher Faciliaion To solve 8, we have been rewriing i in logarihmic form: log 8. Ask sudens wha hey relaionship hey noice beween he base of he power and he base of he logarihm. Coninue by asking wha log is. Poin ou ha insead of remembering in which order o rewrie eponenial equaions as logarihmic equaions, we could insead do he following: 8 log log 8 The log base undoes he base of in he eponenial epression, so ha only is lef. Remind sudens ha whaever hey do o he lef side of he equaion, hey also have o do o he righ. Then finish up he eample using he change of base propery. Logarihms 6

7 log 8 log8 log Someimes, anoher way o solve an equaion of his naure is o ge he same base on boh sides of he equaion. In his case, 8 so we can rewrie he lef side: 8 Now we can ake he log base of boh sides very easily: log log Use hese pracice problems o reinforce he idea of inverse operaions:. Alernae mehod: log5 5 log5 + log5 + log log5 5 + log 5 5. log 7 log Logarihms 7

8 (6 ) 7 65 log (6 ) log65 log log ln + ln f + f ln ln f ln e f ln f e e e f e f 6 6 e e f Suden Applicaion Play he Basic Eponenial and Logarihmic Equaions Round Table Game, which pracices eponenial and logarihmic equaions and he Laws of Logarihms in Logarihmic Equaions Coach/Player Game, which pracices more difficul equaions ha also involve he properies of logs. Embedded Assessmen In he Basic Eponenial and Logarihmic Equaions Round Table Game, sudens check each oher s work and help each oher. In he Laws of Logarihms in Logarihmic Equaions Coach/Player Game, sudens have o verbalize wha hey are seps hey Logarihms 8

9 are aking and help each oher. Boh of hese aciviies can be colleced by he eacher o assess suden undersanding. Reeaching/Eension Use Solving Eponenial and Logarihmic Equaions ha gradually increases in difficuly for sudens who need reeaching. Use he Applicaions of Logarihms for sudens who undersand he concep o pracice real world applicaion of logarihmic and eponenial equaions. Summaive Assessmen: An assessmen of his uni should include problems involving he properies of logarihms and solving logarihmic and eponenial equaions. The assessmen could include brief consruced response or eended consruced response iems similar o he problems in Advanced Pracice. In hese quesions, sudens could be asked o analyze or jusify heir resuls. This uni did no eplicily discuss eponenial growh or decay, which are opics ha could be covered eiher before or afer his uni. Auhors: Name Lana Cone School Garrison Fores School Name Miriam Snare School Mercy High School Logarihms 9

10 Concenraion game wih logarihms. Rules of he game: Sudens should be in groups of players. The deck is shuffled and all cards are laid ou on he able face down. The firs player urns over wo cards. If hey are a maching pair of logarihmic and eponenial forms, he player keeps he pair. If he cards do no mach, he player urns hem back over and leaves hem on he able. If he firs player makes a mach, s/he may ake anoher urn. When no mach is made, i is he ne players urn. Sudens should be observing and checking each ohers pairs o ensure ha correc maches are made. They can coach each oher in he process. Alernaive play: All cards can begin face up and sudens alernae picking up a maching pair. Coaching can occur beween sudens. If he game begins wih all cards face down, sudens could leave he cards ha have been viewed face up for easier maching. Values for Cards: Log7 log5 5 log log log log 7 e log 5 e 5 5 log 8 log log 0 7 log log log 6 7 log 8 6 log Logarihms 0

11 Basic Logarihms Flashcards The difficuly of quesions increases o gradually develop paerns and increase sudens undersanding. wha? wha? wha? 8 Log Log8 Log Log Log 7 Log 9 Log Log 0 Log Log 8 Log Log wha? wha? wha? 8 Log Log Log8 Log Log Log Log 7 Log 7 Log 7 Log9 Log9 Log 7 9 Logarihms

12 Advanced Logarihms Card Game *The difficuly of he problems is adjused for sudens who feel comforable wih he basics. Rules of he game: players in a group. The deck is shuffled and each player is deal 5 cards from he deck. The deck is hen placed face down on he able and he op card is flipped face up. The firs player ries o mach he value on he open card wih a card in heir hand. If hey have a mach, hey draw a card from he deck and flip he op card in he deck face up. If here is no mach, ne player akes he urn. If nobody has a mach, he ne card in he deck is flipped. When all he cards in he deck have been flipped over, urn he enire deck face down again and coninue. The game coninues unil all he cards on he able are mached. Whoever has he mos mached pairs by he end of he game, wins. Alernaive play: Same iniial seup. When he op card in he deck is flipped, he player who maches he card in heir hand he fases akes he open card. A mach can be a numerical value on he card, or a logarihmic epression of he same value. Quesions and Answers for Cards: Log 0 Log 0 Log 0 Log 00 0 Log 00 Log 8 Log 6 Log Log8 Log 5 Log Log 6 Log 7 Log87 0 Log ( ) DNE DNE Log0 DNE Log Log Log 8 Log Log Log 8 Log 7 Log 8 Logarihms

13 Concenric Circles Game Rules of he game: Each suden is deal one card. Sudens are divided ino wo groups of he same size. One group forms he inner circle, he oher forms he ouer circle, and pairs of players from inner and ouer circle face each oher. Each suden in a pair akes urn o hold up heir card for he oher player o see. The oher player mus figure ou he answer which is wrien on he back of he card. If a player gives he correc answer, hey earn a poin. If no, he wo sudens in he pair figure ou he soluion ogeher, and nobody earns a poin. Once each pair has solved heir problems, he inner circle moves wo ouer players o he righ. On he ne urn, he ouer circle moves wo inner players o he righ, ec. Se : Common and Naural logs; Change of Base. Quesions and answers for cards: Log 00 Log 0 Log Log 0 0 Ln e Ln e Ln e Ln 0 Log 0 Ln e Use Base 0: Log? Log.565 Log Use Base 0: Log 9? 5 Log 9.09 Log 5 Use Base e: Log Ln.565 Ln Use Base e: Log 9 5 Ln 9.09 Ln 5 Logarihms

14 Se : Laws of Logarihms. Quesions and answers for cards: Epand: Log mn Answer: Log m + Log n Epand: mn Log p Answer: Log m + Log n Log p Epand: Log m n Answer: Log m + Log n Epand: m Log n Answer: Logm Logn Wrie a Single log Log m Log n Log p + Answer: mn Log p Wrie a Single log Log m + Log n Answer: m Log n Wrie a Single log Answer: Logm Logn + Log mn Epand: Answer: Log 8mn + Logm + Logn Wrie a Single log Log m Log n Epand: 9 Log mn Answer: m Log n Answer: Log m Log n Wrie a Single log Log m + Log n Answer: m Log n Wrie a Single log Log n Answer: 0 Log n Epand: m Log 0n Answer: Logm Logn Wrie as a Single log Log5p Log5q Answer: p Log5 q Logarihms

15 Reeaching Properies of Logarihms Name: Rewrie each problem using he change of base formula and hen evaluae using a calculaor. ) Change o log. log 0 ) Change o ln. log55 ) Change o log. log6 Rewrie each problem using he produc propery. ) Epand. log 5 5) Epand. ln y 6) Condense. log + log p + log Rewrie each problem using he quoien propery. 7 7) Epand. log 8) Epand. ln m 9) Condense. log r log log u Rewrie each problem using he power propery. 0) Epand. log ) Epand. ln y 5 ) Condense. 0log p Logarihms 5

16 Reeaching Properies of Logarihms Name: _ANSWER KEY Rewrie each problem using he change of base formula and hen evaluae using a calculaor. log 0 ) Change o log. log 0 Soluion:.5 log ) Change o ln. log 8 Soluion: ln 8 ln ) Change o log. log6 Soluion: log.5 log6 Rewrie each problem using he produc propery. ) Epand. log 5 Soluion: log 5 + log 5) Epand. ln y Soluion: ln + ln y 6) Condense. log + log p + log Soluion: log p Rewrie each problem using he quoien propery. 7 7) Epand. log Soluion: log 7 log 8) Epand. ln Soluion: ln ln m m 9) Condense. log r (log + log u) Soluion: log r u Rewrie each problem using he power propery. 0) Epand. log Soluion: log ) Epand. 5 ln y Soluion: 5ln y ) Condense. 0log p Soluion: 0 log p Logarihms 6

17 Laws of Logarihms Workshee Name: ) Epand: ) Epand: Log 8mn Log mn ) Epand: Log 0m n 5) Wrie a Single log Ln m + Ln n Ln p Ln q 7) Wrie a Single log Logm+ Logn 9) Wrie a Single log Log n ) Epand: 8m Log n ) Epand: m Log n 6) Wrie a Single log Log m Log n Log p 8) Wrie a Single log + Log m 0) Wrie a Single log Log5k ) Wrie as a Single log 5 Logm Logn + Logarihms 7

18 Laws of Logarihms Workshee Name: ANSWER KEY ) + Logm + Logn ) Logm Logn ) + Log m + Log n ) Log m Log n 5) mn Log pq 6) m Log n p 7) Log m n 8) Log m 7 5 9) Log 0) Log n 5 k ) 5 Log m Log n + ) m Log n Logarihms 8

19 Basic Eponenial and Logarihmic Equaions Round Table Game. Name: Rules of he game: Each suden in a group ( 5 people) receives he workshee. They all sar by working hrough he firs problem and hen pass heir workshee o he ne person clockwise. The ne person checks he previous problem and does he ne one, hen passes he workshee on. The process coninues unil all he problems on he workshees are compleed. ) 9 ) log Check: Check: ) 8 ) log ( ) Check: Check: 5) 8 6) log 7) 5 5 Check: 8) log 5( ) Check: 9) 9 7 Check: 0) log 9( + ) Check: ) ( ) Check: ) log ( ) 0 Check: Check: Check: Logarihms 9

20 Basic Eponenial and Logarihmic Equaions ANSWER KEY ) ) 8 ) ) 0 5) 6) 6 7) 9) 0) 8) 5 ) ) Logarihms 0

21 Laws of Logarihms in Logarihmic Equaions Coach and Player game. Name: Rules of he game: Sudens are paired up, and each pair receives a workshee. Sudens in each pair ake urns assuming Coach and Player roles. The game can ake wo pahs: he Coach can observe and guide as he Player is solving he problem; or he Coach can solve a problem verbally while he Player wries down eacly wha he Coach ells hem o wrie (ecellen way o develop sudens mah vocabulary and ariculaion of ideas). Encourage sudens o use posiive reinforcemen and consrucive bu posiive correcions. ) log + log log 0 ) log8 + log8 ) ln ln ) ln( ) + ln 5 0 5) log log + log 0 6) log log6 Logarihms

22 Laws of Logarihms in Logarihmic Equaions ANSWER KEY ) 5 ) ) 9e ) 6 5 5) 6 6) 6 Logarihms

23 Solving Eponenial and Logarihmic Equaions Name: Solve each equaion. ) 7 56 Think: How do you undo he 7? Think: Wha happens o he? Check! 5 ) 0 Think: How do you undo he? Think: Wha happens o he 5? Check! ) log y Think: How do you undo he log? Think: Wha happens o he y? Check! ) log ( + 5) 5 Think: How do you undo he log? Think: Wha happens o he + 5? Check! Logarihms

24 Solving Eponenial and Logarihmic Equaions Name: _ANSWER KEY Solve each equaion. ) 7 56 log7 7 log7 56 log log 7 ) 5 0 log 5 log0 5 log ) log y log y y 6 log0 ) log ( + 5) 5 log (+ 5) Logarihms

25 Applicaions of Logarihms Name: ) The magniude of an earhquake was defined in 95 by Charles Richer by he I epression M log where I is he inensiy of he earhquake and S is he S inensiy of a regular earhquake. The magniude of a regular earhquake is S M log log 0. S a. If he inensiy of an earhquake in Logarihville is imes he inensiy of a regular earhquake, wha is is magniude? b. Early in he 0 h cenury, he earhquake in San Francisco regisered 8. on he Richer scale. By wha facor of S was he inensiy of his earhquake higher han ha of a regular earhquake? c. In he same year, anoher earhquake in Souh America was recorded o be imes more inense han ha of San Francisco. Wha was is magniude? ) The populaion growh problems are ofen modeled by he equaion N () N( +r) where N 0 0 original populaion size, N( ) final populaion size, r rae of increase/decrease of he populaion, and ime ha has passed. Suppose a group of rabbis increases heir numbers by 0% every year. a. How many rabbis will here be afer years? b. How many years will i ake for he populaion o double? ) Radioacive decay is ofen modeled by he equaion A () A 0 where A 0 he original amoun of elemen presen, A( ) he final amoun, h half life of he elemen (how long i akes for he elemen o decay o half of is original amoun), and ime ha has passed. The radioacive subsance Carbon, presen in all living hings, has a half life of 570 years. By measuring he amoun of C presen in a fossil, scieniss can esimae how old he fossil is. Suppose a group of archaeologiss discovered wha hey hink may be Barny s grea ancesor, Barnyssaorus Re. There seems o be 65% of he radioacive elemen remaining in he bones. How old is he fossil? h Logarihms 5

26 Applicaions of Logarihms Name: _ANSWER KEY ) a. Le he inensiy of he earhquake in Logarihville be I S. Then is magniude, M log I log S log S S S log S b. Le facor of S. Then he inensiy I S, and so I S S M log log log S S log 8. S 8. So log 8. and herefore, 0. I c. M. log SanFrancisco SanFrancisco 8 S ISanFrancisco ISanFrancisco M SouhAmerica log log S S I SanFrancisco log + log S ) a. For his problem, N 0, r 0., and. We wan N ( ).Then N ( ) ( + 0. ) 99. 6, or abou 9 rabbis. b. If he populaion doubles, hen N ( ) N 68. Now is unknown. So, 68 ( + 0. ). Dividing boh sides by yields.. Taking log of boh sides gives us log log. log., and so log. log. 06 years. 0 ) In his problem, A 0 is unknown, bu since we know ha 65% of he elemen is remaining, we can use A( ) 065. A. Half life is h 570, and is unknown. So we ge. A A A 0 A Taking log of boh sides yields log 065. log log 570 log 065. log 065. Solving for, we ge and so years. 570 log log Logarihms 6

27 Summaive Assessmen Name: Rewrie he logarihm in eponenial form. log 5 ) 5 ) log8 m Rewrie he eponen in logarihmic form. ) 6 m ) Evaluae he logarihms using log 0.60 and log.6. 5) log 6) log 7) l og56 8) l og6 Logarihms 7

28 Solve he equaion. 9) 8 w ) 5 00 ) ln 6 + ln ) log log 7 Logarihms 8

29 ) Maryville was founded in 950. A ha ime, 500 people lived in he own. The populaion growh in Maryville follows he equaion P , where is he number of years since 950. a) Deermine when he populaion had doubled since he founding. b) In wha year was he populaion prediced o reach 5,000 people? c) Wha social implicaions could he populaion growh in ha number of years have on he own? Logarihms 9

30 Summaive Assessmen Name: Answer Key Rewrie he logarihm in eponenial form. ) 5 5 ) m 8 Rewrie he eponen in logarihmic form. ) log 6 ) log m Evaluae he logarihms using log 0.60 and log.6. 5) log.6.90 log.60 6) l og log ) l og log + log ) log log Solve he equaion. 9) w w 8 0 log (8 ) log 0 w 8 8 log 0 w log8 0. log8 0) 5 00 log ( ) log 00 5 log00 5 log00 log Logarihms 0

31 ) ) ln 6 + ln ln e e ln log log 7 log 7 log ) Maryville was founded in 950. A ha ime, 500 people lived in he own. The populaion growh in Maryville follows he equaion P , where is he number of years since 950. a) Deermine when he populaion had doubled since he founding. P log The populaion had doubled in he year 965. b) In wha year was he populaion prediced o reach 5,000 people? log log The populaion had reach 5,000 jus prior o he 975. c) Discuss wheher he populaion growh prediced by his equaion is reasonable over a enire cenury. Include any social implicaion ha could occur. Logarihms

32 Suden answers should include comparing he original populaion and he populaion in 975 or laer. Discussions could include job and housing availabiliy, raffic implicaions of populaion boom, ec. Logarihms