Advanced Functions Page 1 of 3 Investigating Exponential Functions y= b x

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1 Advnced Functions Pge of Investigting Eponentil Functions = b Emple : Write n Eqution to Fit Dt Write n eqution to fit the dt in the tble of vlues Properties of the Eponentil Function =b () The bse b is positive. () The -intercept=. () The horizontl smptote is the -is (i.e. =0) (The grph does not touch the -is, but getting close to it.) (4) The domin: R. () The rnge: R, >0 (i.e. the H.A.) () If b>, the eponentil function is lws incresing. (7) If 0<b<, the eponentil function is lws decresing. Note: () ; () The grphs =b nd =b - re smmetricl bout the -is. = - = - = = Emple Grph nd. ) Wht trnsformtion on will give b) How re the curves like nd different? s its imge?

2 Advnced Functions Pge of Investigting Eponentil Functions = b Emple Grph nd. ) Wht trnsformtion on will give b) Wht trnsformtion on will give c) How re the curves like nd different? s its imge? s its imge? Emple 4: Grphing n Inverse Function Consider the function f ( ) 4 ) Identif the ke fetures of the function. i) domin nd rnge ii) -intercept, if it eists iii) -intercept, if it eists iv) intervls for which the function is positive nd intervls for which it is negtive v) intervls for which the function is incresing nd intervls for which it is decresing vi) eqution of the smptote b) Sketch grph of the function. c) On the sme set of es, sketch grph of the inverse of the function. d) Identif the ke fetures, s in prt ) i) to vi), of the inverse of the function Homework: P. 8 #,,,,7,9,-

3 Advnced Functions Pge of Investigting Eponentil Functions = b Steps of sketching the grph =b +c () Use tble to drw the bsic grph =b. () Find the -intercept. () Find nd drw the horizontl smptote. (4) Trnslte =b pssing through the -intercept nd close to the horizontl smptote. Emple In ech given eponentil function below, find i) the bse ii) the -intercept iii) the horizontl smptote iv) the domin v) the rnge vi) the increse nd decrese of the function. vii) sketch the function () (b) Homework. #-

4 Advnced Functions Pge of Solving Eponentil Equtions Recll () () () m m n mn (4) n mn () m n mn ( ) () m m m ( b) b (7) m b b m m m m b m (8) 0 n m b m (9) n m Emple Evlute ech of the following: 7 () (b) 4 (c) 8 4 Emple Simplif ech of the following, using the lws of eponents. (b) 0 () 4 (c)

5 Advnced Functions Pge of Solving Eponentil Equtions Emple : Simplif Eponentil Epressions Simplif the following Eponentil Epression m m m m m 4 Emple 4: Solving Eponentil Equtions Solve the following eponentil equtions: ) b) 4 c) 0 d) 8 Emple If m nd k re integers, find ll solutions to the eqution k k m m 9(7 7 ) Homework: P.0 #dgjkl, def, 4dfh, ef, b, 7 WS: Eponentil Questions

6 Advnced Functions Pge of Solving Eponentil Equtions ) Simplif ech of the following. n n n 4 8 ) b) n n 7 n n 9 n n c) n n n 9 d) 0 0 n n 4 n n e) p. p p p f) p p p ) Solve ech of the following eponentil eqution. ) 8 0 b) 0 0. c) 7 7 d) e) 0 4 f) g) h) j) 80 i) 7 9 ) F is n eponentil function such tht F b. If F 00 nd 000 F, find nd b. 4) Solve ech of the following pirs of equtions. ) 4 b) ) Simplif ech of the following. ) n n4n n n9 n b) z z z z z z Answers ) n b) n c) ) b) 0 or c) 0. n p d) e) 9 n p d) - or e) f) p or f) 8 or 7 g) h) 0 or i) j) 4 ), or, 4), or, b) (, ) or, ) b) z

7 Advnced Functions Pge of 4 Eponentil Growth nd Dec The eponentil function occurs frequentl in nture nd societ. For emple, it cn be used in problems of popultion growth, compound interest, investment growth, deprecition, infltion rtes bcteril growth, nd rdioctive dec nd so on. Growth t t ( r) R, where R t o t o (This formul is used to find "rte") 0 = originl vlue r = growth % t = vlue fter time t R = (+r) Dec t t ( r) R, where 0<R< t o t o (This formul is used to find "rte") Emple An ntique vse ws purchsed in 00 for $9000. If the vse pprecites in vlue b % per er, wht is its estimted vlue in the er 00, to the nerest thousnd dollrs? Doubling Period (d) It is the time to mke n object double its vlue. () t o t d Emple The popultion of the world ws billion in 999. This popultion is growing eponentill nd doubles ever ers. () Estimte the world popultion in 040, to the nerest hlf billion. (b) When will the popultion be 48 billion?

8 Advnced Functions Pge of 4 Eponentil Growth nd Dec Emple The popultion of town ws 4000 in 980 nd 9000 in 990. () Determine n epression for the popultion t the time t ers fter 980. (b) Use this epression to estimte the popultion of the town in 00. Emple 4 Just s popultion growth nd infltion cn be described b n eponentil growth function, rdioctive dec nd deprecition cn be described b n eponentil dec function. A cr deprecites b % per er. If ou bu cr for $000, find the vlue of the cr in three ers. Hlf-life (h) It is the time to reduce the mss of n object to its hlf. (0.) Emple An isotope of rdium is used b hospitl for cncer rdition. The hlf-life of this rdium is 0 ers. If the hospitl initill hd 0 mg, how much will the hve fter 0 ers? t o t h

9 Advnced Functions Pge of 4 Eponentil Growth nd Dec In generl: Eponentil Growth nd Dec For problems of growth, dec, doubling period, nd hlf-life, the following formul cn be used for ll of them. t o e kt o = the originl vlue t the vlue fter time t t = the time The constnt k must be found first. To find the rte mens tking the derivtive (usull with respect to t). Emple : The count in bcteri culture ws 000 fter min nd fter h. ) Wht ws the initil size of the culture? b) Find the popultion fter t hours. c) When will the size of the popultion be 0000? ) Recll : i) ii) ln e ln b ln e ln b b Emple 7: An isotope of sodium hs hlf-life of h. A smple of this isotope hs mss of g. ) Find the mss tht remins fter t hours. b) Find the mss tht remins fter h. Homework WS: Eponentil Growth nd Dec Eercise P. # -

10 Advnced Functions Pge 4 of 4 Eponentil Growth nd Dec ) The popultion of cit is If it is incresing b 4% per er, estimte the popultion in 4 ers? ) A pinting, purchsed for $0 000 in 990, incresed in vlue b 8% per er. Find the vlue of the pinting in the er 000. ) A river is stocked with 000 slmon. The popultion of slmon increses b 7% per er. ) Write n epression for the popultion t ers fter the slmon were put into the river. b) Wht will the popultion be in i) ers? ii) ers? c) How mn ers does it tke for the slmon popultion to double? 4) A house ws bought si ers go for $ If rel-estte vlues hve been incresing t the rte of 4% per er, wht is the vlue of the house now? ) A used-cr deler sells five-er old cr for $400. Wht ws the originl vlue of the cr if the deprecition is % er? ) In the erl 990s, the Cndin dollr ws declining in vlue due to infltion t the rte of 8.% per er. If the sitution continued, wht would the dollr be worth five ers lter? 7) To determine whether pncres is functioning normll, trcer de is injected. A normll functioning pncres secretes 4% of the de ech minute. A doctor injects 0.0 g of the de. Twent minutes lter, 0. g remin. If the pncres were functioning normll, how much de should remin? 8) If bcteri popultion doubles in d, ) when will it be times s lrge? b) when ws it hlf of its present popultion? c) when ws it of its present popultion? 4 d) when ws it of its present popultion? 9) Infltion is cusing things to cost roughl % more per er. ) A bg of milk costs $.7 now. Estimte its cost in five ers. b) i) A movie ticket costs $8.0 now. If infltion continues t % /er, when will the ticket cost $0? c) ii) How long go did the movie ticket cost $4.? 0) An element is decing t the rte of %/h. Initill we hve 00 g. ) How much remins fter 0 h? b) How much remins fter 0 h? c) When will there be 40 g left? ) A reserch ssistnt mde 0 mg of rdioctive sodium (NA 4 ) nd found tht there ws onl 0 mg left 4 h lter. ) Wht is the hlf-life of NA 4? b) Find function tht models the mount A left fter t hours. c) If the lbortor requires 00 mg of NA 4 h from now, how much NA 4 should the reserch ssistnt mke now? (Ignore the 0 mg she currentl hs.) d) How much of the originl 0 mg would be left in h? ) A bcteri colon grows t the rte of %/h. ) In how mn hours will the colon double in size? b) In 0 h the bcteri popultion grows to. 0. How mn bcteri were there initill? Answers ) ) $00 ) P b) i) ii) 800 c) 0. ers 4) $000 ) $900 ) $0. 7) 0.g 8) 0 ds b) ds go c) 0 ds go d) ds go 9) $4.4 b) i) 8 ers ii) ers go 0) 8g b) g c) 7 h ) h b) t A 0 c) 74 mg d). mg ) h b) 0

11 Advnced Functions Pge of The Logrithmic Functions Logrithmic Function vs Eponentil Function To chnge the eponentil function to rithmic function, or vice vers, use the fct tht: b. e.g. b Common Logrithm The common rithm is the rithm with bse s 0. This bse 0 is usull omitted ; e.g Emple Chnge to rithmic form () 7 9 (b) 8 Emple Chnge to eponentil form () 9 (b) 9 7 Emple 4 Use our clcultor to find the vlue of the following: () 0 00 (b) 0.0

12 Advnced Functions Pge of The Logrithmic Functions Bsic Properties of Logrithm ) 0 ) ) 4) b ) b ) (Chnge of Bse) Emple Evlute () 8 (b) 8 (c) 4 Emple Solve for () (b) 7 (c) Emple 7 If ( ), find. 4 Emple 8 Solve the equtions for the point of intersection of the grphs of = () nd = 4. Homework P. 8 #-4, 8, 0, -

13 Advnced Functions Pge of Lws of Logrithms Lw Generl Rule Proofs Emples: Use the lws to epnd the following single rithms Product 7 Quotient 00 n Power n 8 Emple Epress s sum nd difference of rithms: z 7 w v 4 7 Emple Epress s single rithm: w 7 m

14 Advnced Functions Pge of Lws of Logrithms Emple Simplif the following: 7 ) ( ) 8 b) 48 c) 4 Emple 4 If = nd =, epress the following in terms of nd. ) b) 4 Emple If 8 =k then epress 8 8 in terms of k. Emple Evlute 49 Emple 7 Solve to two decimls. ) b) 7(.0 ).0 Homework Tetbook: Power Lw: P. 47 #-,9,0,,0 Product & Quotient P. 84 #-7,9,0

15 Advnced Functions Pge of Sketching Logrithmic Functions Recll: Logrithmic Functions Emple Sketch ech rithmic function nd describe its trnsformtion. ) c) The bse is positive. The -intercept=. Verticl smptote is the -is (i.e. =0). Domin:, 0. & Rnge: If b>, the function is incresing. If 0<b<, the function is decresing. b)

16 Advnced Functions Pge of Sketching Logrithmic Functions e) d) f) 4 Homework: Sketching Logrithmic Functions

17 Advnced Functions Pge of Sketching Logrithmic Functions For grphs # to #, first grph the bsic eponentil function which corresponds to the given rithmic function, then grph the rithmic function s reflection in of the eponentil function. For emple, if sked to grph : first grph then grph its reflection in, tht is : Since grphs #4 to # re ll bsed on grph #, it is not necessr to sketch the required eponentil function for these. Just refer bck to our work in grph # ( )

18 Advnced Functions Pge 4 of Sketching Logrithmic Functions (Hint: use lw #.). ( ) (Hint: tr both positive nd negtive vlues of mke tble of vlues nd grph from this.)

19 Advnced Functions Pge of Sketching Logrithmic Functions Answers ( )

20 Advnced Functions Pge of Sketching Logrithmic Functions 0.. (Hint: use lw #.). ( ) 4.. (Hint: tr both positive nd negtive vlues of mke tble of vlues nd grph from this.)

21 Advnced Functions Pge of The Nturl Eponentil & Logrithmic Functions Definition of e (Nturl Eponentil Number) e 0 lim( ) e Properties of e nd ln Recll the rithmic function is the inverse of the eponentil function. b which is theinverse of b, b e e which is theinverse of e. The function e cn be written s ln nd is clled the nturl rithm function. b b ln e e e ln Domin is R Rnge is R 0 -intercept t ln e, 0 Horizontl Asmptote 0. Domin is R 0 Rnge is R -intercept t ln e, R Verticl Asmptote 0. = e ln ln ln ln ln Bsic Properties ) 0 ln e 0 ) e e= ln e ) e ln e, R e e 4) e e ln, > 0 Definition of Nturl Logrithm Nturl rithm is the rithm to the bse e. ln Lws of Logrithm ) ln ln ln (Product Lw) e e e ) e e e ln ln ln (Quotient Lw) p ) p ln p pln (Power Lw) e e e Emple : Nturl Logrithms Solve for : ln 8

22 Advnced Functions Pge of The Nturl Eponentil & Logrithmic Functions Emple : Solving Eponentil Equtions Solve the eqution e 4 Emple : Nturl Logrithmic Function f ln 4. Find the domin of the function Emple 4: Single Nturl Logrithm Epress ln 4ln ln s single rithm. Eponentil Growth & Dec If f t is the number of individuls in popultion of nimls or bcteri cells t time t, then the size of the popultion t time t is kt 0e.

23 Advnced Functions Pge of The Nturl Eponentil & Logrithmic Functions Emple : Eponentil Growth A bcteri culture strts with 000 bcteri nd fter h the estimtes count is bcteri. ) Find the number of bcteri fter t hours. b) Find the number of bcteri fter hours. c) When will the bcteri popultion rech 8 000? Emple : Eponentil Growth The hlf-life of Polonium-0 is 40 ds nd smple of this element hs mss of 00 mg. ) Find the mss tht remins fter t ds. b) Find the mss tht remins fter 0 ds. c) How long will the smple tke to dec to mss of 00 mg? HW: Worksheet Nturl Logrithmic Eercise

24 Advnced Functions Pge 4 of The Nturl Eponentil & Logrithmic Functions Eercise. Grph ech function, stte the domin, rnge, nd smptote of ech function. ) g ln b) ln c) ln d) ln e) ln. Evlute without using clcultor. ln ln ) e b) lne c) lne d) e e) ln e f) ln ln ln8. Solve for : ) e 4 b) ln c) ln d) 0 f) ln ln4 ln7 g) lnln h) e e e e) lne 8 4. Find the solution of ech eqution correct to si deciml plces. ) ln b) e c) e 0 d). Epress s single rithm. ) ln ln b) ln ln ln. Find the domin of ech function. g ln b) h ln ln c) h ln0 ) 7. A bcteri culture strts with 000 bcteri. After h the estimted count is 0000 bcteri. ) Find the number of bcteri fter t hours. b) Find the number of bcteri fter h. c) When will the bcteri popultion rech 000? 8. The initil size of bcteri culture is 400. After n hour there re 00 bcteri. ) Find the number of bcteri fter t hours. b) In wht period of time does the popultion double? 9. A cell of the bcterium Escherichi coli in nutrient broth medium divides into two cells ever 0 min. Suppose tht there re initill 00 cells. Find ) the number of cells fter t hours. b) the number of cells fter 8 hours. c) the time required for the size to rech 000 cells. 0. The popultion of the world is doubling bout ever ers. In 987 the popultion reched billion. ) Find the projected world popultion for the er 00. b) When will the world popultion rech 0 billion? 9. Urnium-8 hs hlf of 4. 0 ers. ) Find the mss tht remins from 00 mg smple fter t ers. b) Find the mss tht remins from this smple fter 0000 ers.

25 Advnced Functions Pge of The Nturl Eponentil & Logrithmic Functions Answers ) D : (, ) R : R, Asmp: b) D : (0, ) R : R, Asmp : 0 c) D : (,0) R : R, Asmp : 0 d) D : (0, ) R : 0,, Asmp : 0 e) D : (,0) 0, R : R, Asmp : 0 ) b) c) d) e) 0. f) 0 ) ln4 b) e e c) 0.(e+) d) ln0 e) - f) 8 g) ln ln 4) b) c) d).849 ) ln b) ln ) (0, ) b) (0, ) c) [, 0) t 7) ) t b) 8 c) hr min. ln 400 b) h 7. mi n ln ln ln t 9) 00 b) c) h hr min 0) 47 billion b) 0 ) t b) mg e h)

26 Advnced Functions Pge of Chnge of Bse () () b b b b b () In prticulr, b b 0 Emple Evlute to deciml plces: ) b) Emple Prove tht t b. t b Emple Show tht. 4

27 Advnced Functions Pge of Chnge of Bse Emple 4 If b 4 b, where 0, b 0, show tht b ( b) 4 Emple P.8 #8 4 If b p nd b, show tht p p. 4 Emple ( )( 4)( )...( ), clculte the vlue of. If 4 Homework: WS: Chnge of Bse

28 Advnced Functions Pge of Chnge of Bse () b b b () b () In prticulr, b b b 0. Use our clcultor to find the vlue of ech of the following, correct to three deciml plces. ) b) 7 4 c). 4 d) Show tht ech of the following sttement is true. ) b) c) d) 8 4. Describe the chnges to the grph of when is replced b 4. For, b, show tht b. If b b, where 0, b 0. For 0,, 0, prove tht. b b., show tht b. 7. If 4 b show tht p b p nd, p If b b b, where 0, b 0, b show tht b. Answers ).89 b).477 c) 0. d). ) Grph is reflected in is.

29 Advnced Functions Pge of 4 Solving Logrithmic Equtions () Mking both sides of the eqution with single of the sme bse. () When ech term of n eqution is rithm of the sme bse, ppl lws of rithm to mke ech side s single rithm. () From (), we hve the form: A B A B () Solve for the unknown in A=B. (4) Check ech root in the originl eqution. Reject n root tht mkes rithm of negtive number. Emple If N z, solve for N in terms of,, nd z. Emple Solve for : 7 Emple Solve for : 4( ) 4( ) 49

30 Advnced Functions Pge of 4 Solving Logrithmic Equtions (b) Solving rithmic equtions with constnt terms () Collect ll the terms of rithms with sme bse on one side, nd mke them s single rithm. () Write the eqution in the form N=number. () Chnge to eponentil form, nd solve the eqution. (4) Check ech root in the originl eqution. Reject n root tht mkes rithm of negtive number. Emple 4 Solve for : 4 4( ) Emple Solve for : 80 ( ) Emple Determine the points of intersection of the curves 0( ) nd 0( ). Emple 7 Eplin wh there re no solutions to the equtions ( ) nd Homework: p.9 #,,, 9, (omit b) 7. #-4, -8

31 Advnced Functions Pge of 4 Solving Logrithmic Equtions ) Solve ech of the following eponentil eqution. ) 8 b) 0 0 c) 0 d) 4 e) ) Solve ech of the following eqution for. ) 0. b) 8 7 c) d) 8 7 e) 4 0. f) ) If nd b, find 0 9 in terms of nd b. 4) Solve the sstem of equtions z z z 0 ) Determine tn tn tn... tn ) If w 4, w 40 nd z w, find w 7) Solve ech of the following eqution for. ) b) c) 0 d) 4 g) z e) 4 f) 4 h) i) 0 j) 0 k) 0 l) 0 0 m)

32 Advnced Functions Pge 4 of 4 Solving Logrithmic Equtions 8) Solve ech of the following to deciml plces. ) 0 b) 4 0 c) 8 9 9) If m, find the vlue of ech of following in terms of m: ) b) c) d) e) f) 0) Simplif ech of the following. ) b) c) d) 4 ============================================================================ Answers: ) 0 & b) c) & - d) & 0 e) & - & - ) b) & c) 7 d) e) 7 f) 900 ) b b 4) 8 7,, z ) 0 ) 0 7) b) c) d) e) 4. f) 4 g) h) No Solution i) No Solution j), k) 4 l) 0 & 0 m) 8) -0.4 b).8 c) & 9) m b) -m c) m d) m e) m f) m 0) 4 b) -7 c) d) 8

33 Advnced Functions Pge of Logrithmic Applictions () Logrithms nd Erthqukes The formul Richter used to define the mgnitude of n erthquke is I M I o I= intensit of the erthquke being mesured; I o = intensit of reference erthquke; M=number used to mesure the intensit of n erthquke. Emple The Alsk erthquke of 94 hd mgnitude of 8. on the Richter scle. How mn times more intense ws the Alsk erthquke thn modertel destructive erthquke which hs mgnitude of.0? Let I A be the intensit of the erthquke in Alsk I M be the intensit of moderte () Logrithm nd Sound The formul used to compre sound is L 0 I I o I=intensit of the sound being mesured; I o intensit of the sound t the threshold of hering L=Loudness of sound mesured in decibels (=/0 of bel).

34 Advnced Functions Pge of Logrithmic Applictions Emple A sound is 000 times more intense thn sound ou cn just her. Wht is the mesure of its loudness in decibels? Emple How mn more times intense is the sound of norml converstion (0 db) thn the sound of whisper (0 db)? () Logrithms nd Chemistr Chemists define the cidit of liquid on H p scle. The formul is p H [ H ] H p =cidit of liquid H =concentrtion of the hdrogen ion in moles/litre.

35 Advnced Functions Pge of Logrithmic Applictions Emple 4 Find the ph of swimming pool with hdrogen ion concentrtion of.0-8 mol/l. Emple The ph of fruit juice is.0. Wht is the hdrogen ion concentrtion of the fruit juice? Homework P. #-4,-, P. 40#,,,7,9 Logrithmic Appliction Review

36 Advnced Functions Pge of Review: Eponentil & Logrithmic Functions ) Evlute ) 9 b) c) d) e) f) g) h) 7 ) Appl the Lws of Logrithms to the following. ) 0 (4 9) b) ( ) c) 0 d) 0 e) f) 0 g) () h) i) 4 4 j) ) Evlute ) b) 4 4 c) f) 9 g).7 00 h) d) e) i) j) 0 4) Solve the following equtions for. ) 8 b) c) 0 0 d) 4 e) f) 4 8 g) 8( ) 8( ) h) 8( ) 8( ) ) Epress s single rithm. ) 0 0 b) c) M N P d) 4 A B ) Find the roots of the following equtions correct to 4 deciml plces. ) b) 4 c). d) 0 8 7) A bcteri culture doubles ever 0 min. After n hour there re 000 bcteri. ) Wht is the initil size of the culture? b) Wht is the size fter t minutes? c) Wht is the size fter h? d) Wht is the size fter h?

37 Advnced Functions Pge of Review: Eponentil & Logrithmic Functions 8) A bcteri culture strts with bcteri. After 40 min the count is Wht is the doubling period? 9) A smple of Rdon- decs to 0% of its originl mss in 9 h. Wht is the hlf-life of Rn? 0) The hlf-life of V 48 is d. ) Find the mount remining from 0 mg smple fter i) t ds; ii) d; iii) 40 d. b) How mn ds will it tke to dec to mg? ) Solve the following equtions. ) ( ) ( ) b) c) 0 ) Find the domin of the function f ( ) 0( ). ) Find the domin of the function g( ) ( ). 4) Determine the eqution for the following grphs, ) b) d) (0 ) Answers ) b) 4 c) d) e) 0 f) - g) d) e) 7 h) 7 ) b) c) 0 f) 0 g) h) i) j) 97 ) - b) c) -7 d) e) f) g).7 h) 97 i) j) 4) 8 b) c) d) 4 e) f) g) h) ) 0 b) ( ) c) ( MN P) d) A 4 ).449 b) 0.84 c) -. d).87 B 7) 4000 b) c) d) t t ).4 min 9).9 h 0) i) 0 mg ii) mg iii).0mg b) 0. d ) 0 4 b) 4) 4( ) b) ( 4) c) ( 0) d) 0 ) ) 0