Name: Period: Pre-Cal AB: Unit 16: Exponential and Logarithmic Functions Monday Tuesday Block Friday. Practice 8/9 15/16. y y. x 5.

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1 Ne: Period: Pre-Cl AB: Uit 6: Epoetil d Logrithic Fuctios Mody Tuesdy Block Fridy 0,, 6, 5, 7 April,, 6,, 7 CONICS DOUBLE QUIZ Rdicl d Rtiol Epoets Prctice HOLIDAY 6 Grphig Solvig Applictios 7 QUIZ Mied Prctice 8/9 5/6 Lesso #: Rdicl d Rtiol Epoets I c Use properties to siplify epoetil epressios. Product of Powers Property: = + Quotiet of Powers Property: = Power of Power Property: ( ) = Power of Product Property: ( b) = b Power of Quotiet Property: b = b Negtive Epoet Property: = Zero Property: 0 = The epoet idictes the th root. The epoet idictes the th root rised to the th power. More Eples = ( ) = OR = E: E: E: E: E: E: E: E: E: Norl Mody schedule Logrithic Properties Solvig Logs d Epoets 5 ( )( ) Uit 6 TEST 0 Ed of weeks d Eteded Advisory Solvig Logs d Epoets y y 5 5 ( ) ( y ) 6 8 y b c b c. 5 5s t t s

2 Prctice # - No Clcultor. Evlute ech epressio ( 5 )( 5 + ) ( )( 7 ). ( ) c d ( c d ) 6 8 i 6. ( 5 + 8)( 7 ) b 5 7 i i ( ) ( ) b ( ) 7. ( 8) ( 9b) i i i i Notes # Logrithic Properties I c Use properties to siplify logrithic epressios. Trslte betwee logriths i y bse. Covert betwee logrithic d epoetil fors I. Bsics A. Epoetil for: b = ; where b is the, is the, d is the vlue. B. Logrithic for: log b = ; b is the, is the, d is the vlue. C. You c red this s log of bse b or log bse b of D. If the bse is ot give (like i log ) it is uderstood to be bse. E. Leohrd Euler (proouced OILER NOT Yuler) worked with this uusul uber [ e coig fro e = li + ] d itroduced the ottio ed fter hiself. Log bse e is writte s L. It is the turl log. II. Coversios A. To logrithic: ke the epoetil BASE the log. Tke the vlue d ove it over s the d ove the epoet to the other side of the equtio. e. 7 = would be writte s: 9 B. To epoetil: ke the log bse the epoetil. Move the rguet to the other side, d brig the vlue over s the. e. log 00 = would be:. II. Logrithic Properties A. Product Property of Logriths: ( ) log b = where,, d b re positive d b

3 b. b. B. Quotiet Property of Logriths: log b = where,, d b re positive d C. Power Property of Logriths: log b = where is rel, d b re positive, d log( ) D. Chge of bse forul: logb = log( b) E. Iverse Properties:. Arguet d bse re equl. log b = b b. log b = b b = logb. III. Specil Properties d How to Evlute. A. Arguet = : logb = 0 sice. B. Vlue of the rguet: > 0. C. Vlue of the bse: b > 0 d b. D. How to evlute log: Covert to epoetil or use chge of bse forul IV. Model Probles Guided Prctice Covert to log for: O Your Ow = 8 Covert to log for: 0 = 00 Covert to epoetil for: log = 6 Covert to epoetil for: log 8 = Evlute w/o clcultor: log Evlute w/o clcultor: log 6 9 Evlute: log 0 Evlute: log ( ) c b Epd fully: logb k+ Epd fully y log b z :

4 Codese: Codese: Usig log 0.0 log 0.77 pproite log Approite 9 log 8 Prctice # Covert fro log for to epoetil or epoetil to log for. ) l = ) l 9 = ) e = 5 ) i e = 5) b = 0) = 6) 0 7 = 7) 0 k = r 8) 5 = 9) 7 = 00 ) log 000 = ) log 0.00 = - ) log5 = ) logb 7 =.09 5) log 7 = c 5 Epd ech logrithic epressio 5 b r 6) l ( y ) 7) log 8) log 5 y 5 Codese ech logrithic epressio. 9) l + l b + l l y 0) logb p logb v + logb c ) log t ( log + log y) Use the iverse properties to siplify ech epressio without clcultor. ) l e ) log0 ) 7 6) logb b 7) log 7 Use the chge of bse forul to evlute ech with clcultor y log 5) l ( e + ) l 0) log 0 ) log9 6 ) log Evlute ech without clcultor ) log 0,000 ) log 9 5) log 8 9) log8 9 0) log 6 ) log8 8 e 8) ( ) 7 9 6) log 00 + log 9) log8 8 7) log7 8) log6 0

5 log log Give, evlute ech logrithic epressio usig properties. log 0.77 log 0.00 ) log 8 ) log 5 ) log 00 5) log 7 Notes # Grphig Epoetils d Logrithic Fuctios I c Grph d idetify trsfortios of epoetil fuctios Grph d idetify trsfortios of logrithic fuctios Trsfor epoetil d logrithic fuctios grphiclly by usig the rules of pret fuctios. I. Pret Grph Rules A. Geerl For: y = ( f ( h)) + k B. is the sclr d you to the vlues. If is egtive, it reflects cross the -is. C. ( h) ove h uits while ( + h) oves h uits. D. + k oves k uits d k oves k uits. II. Epoetil d Logrithic Fuctios A. Move your origi to (h, k). B. Epoetil poits,, reeber for e use d for /e use C. Sice logs re of epoetil, we plot,, d B. You c grph epoetils like y = e usig e to be roud.7 d /e bout III. Model Probles Guided Prctice Grph y = Grph O Your Ow y = e + Grph y = l( + ) + Grph y = 5 log

6 Grph y = e Grph y = l( ) + y = Grph stte the doi, rge d y = log ( ) the equtios of the syptotes. Grph y = 5 stte the doi, rge d y = log ( ) 5 the equtios of the syptotes. Write equtio tht trsltes f() = log, 6 uits left, uits dow, d hs verticl copressio of ½. Write equtio tht trsltes h() = l, uit left, 6 uits up, d hs verticl stretch of. Describe how the grph is trsfored: + f ( ) = ( ) 8 5 Describe how the grph is trsfored: g( ) = ( )

7 Prctice #- Grph ech equtio sttig the doi, rge, d syptote d otig the criticl poits. ) y = l( + ) ) y = log + ) y = log ( + ) ) y = 5) y = e 6) y = y = + 8) 7) ( ) y 6 = 9) y ( ) = log + Notes #: Solvig I c Solve epoetil d logrithic equtios. Solve word proble pplictios tht ivolve epoetil d logrithic equtios. I. Solvig Logrithic Equtios A. Isolte the logrith if there is oly i the equtio. B. If there re ore th oe, get the o the the use the properties to codese to sigle logrith. C. Covert to epoetil: log b = D. You ust check to ke sure tht your swer does ot ke

8 II. Solvig Epoetil Equtios A. See if you c get ll of the bses to be. If you c, seprte the d solve. Eple = 7 B. If you c t d oly hve oe epoet, it, the covert to. b = C. If there re ore th oe epoet d they hve bses tht cot be de the se, seprte the, epd usig properties of logs, distribute d solve. III. Model Probles Guided Prctice Solve = Solve 9 = 7 8 O Your Ow Solve = 5 Solve 5 = 00 Solve log 6( ) = Solve log00 log ( + ) = Solve log + log ( + ) = Solve log ( 5) = Solve log5 8 = Solve log + log ( + 9) = Solve + = Solve log( + ) = 5

9 Solve 7 e = Siplify 6l e Solve l( ) + l( + ) = l Solve l e = 6 Prctice # - Solve ech equtio to decil plces if ecessry. ) e = e ) ( ) = ) e + 5 = 60 = 5) c 5 ) ( ) e e + = 0 6) l = 7) 5 + l = 8) log 5( ) = 9) log(5 ) + log( ) = 000 0) + e = ) = 80 ) 7 = + ) = ) l( + 5) = l( ) l( + ) 5) 6 log (0.5 ) = 6) ( ) 6 7 = 9 7) + e = 8) l + = 9) 00 = 50 + e 9t.7 0) = 0 log + log ( + ) = log ( + 6) ) log + log ( 8) = ) ) e e = 0 ) e 5e + 6 = 0 5) 5( ) 8 =

10 Notes #5: Applictios I c Use the give foruls to solve word probles. I. Foruls t A. Hlf-Life: N = N h 0 where N =, N 0 =, = tie, d = hlf-life. This fuctio is epoetil d so it decreses i vlue. T T T T e B. Newto s Lw of Coolig: ( ) T 0 =, k =, d t =. kt = R + 0 R where T =, T R =, t r C. Copouded Iterest: A = P + where A =, P =, r =, =, t =. D. Copouded Cotiuously:, t =. A rt = Pe where A =, P =, r = rt E. Epoetil Growth: N = N0e with N0 beig F. f ( ) = P( ± r) II. How to Solve A. If you re NOT lookig for epoet the you c just plug everythig i d solve the equtio. B. If you ARE lookig for epoet the you will plug i the vlues, siplify where possible, the to logrithic for. C. As lwys, if there is the uber, the you will use l ( ). If LTsh deposited $5000 ito ccout tht pys 5% ul iterest copouded cotiuously, fid how y yers it would tke rt for her to hve t lest $0,000. Use A = Pe The hlf-life of plutoiu is bout,0 yers. If grs were relesed ito the eviroet, how y grs would be left fter 000 yers if the decy costt is bout d usig N ( t) = N e kt (just like the bkig equtio) 0

11 A hrd-boiled egg t teperture 96 o C is plced i 6 o C wter to cool. Four iutes lter the teperture of the egg is 5 o C. Use Newto s Lw of Coolig to deterie whe the egg will be 0 o C. Suppose you deposited $50 ito ccout tht pys.5% iterest copouded qurterly. How log will it tke for your ccout to double? ( ) T( t) = T + T T e 0 T 0 = 96, T = 6, T(t) = 5, d t = Fid k. The use k d T(t) = 0 to fid how log you hve to wit. kt r Prctice #5 A = P + t A rt = Pe ( ) (0.5) h rt f = P N = N0e f ( ) = P( ± r) ) Usig 0 th cetury US cesus dt, the popultio of New York Stte c be odeled by: P( t) =, t e where P is the popultio i illios d t is the uber of yers sice 800. Bsed o this odel, A) Wht ws the popultio of i 850? B) Wht will be the popultio i 00? ) The uber B of bcteri i Petri dish culture fter t hours is give by: popultio? B) Whe will the popultio rech 6,000? B 0.69t = 00e. A) Wht ws the iitil t ) The out C i grs of crbo- preset i certi substce fter t yers is give by: C = 0e. How uch of the iitil 0 grs will be left fter 0,00 yers? ) Atscocit High School hs 00 studets i Blue d Gold house. Or, D, Ry, d Le strt ruor which 00 spreds logisticlly so tht S( t) = odels the uber of studets who hve herd the ruor by the ed of t 0.9t + 9e dys where t = 0 is the dy the ruor beg to spred. How log does it tke for 000 studets to her the ruor? 5) The hlf-life of certi rdioctive substce is 65 dys. There re.5 grs preset iitilly. Whe will there be less th gr reiig? 6) If Jose ivests $00 i svigs ccout with 9% iterest rte copouded qurterly, how log will it tke util Jose s ccout hs blce of $50?

12 7) How log will it tke Lure to double her deposit of $8,000 if the ccout she puts the oey i copouds her iterest cotiuously t 6.%? 8) If Joeit ivests $500 ito retireet ccout with 8% iterest rte copouded qurterly, how log will it tke util the ccout grows to $9,000? 9) A totl of $,000 is ivested t ul rte of 9%. Fid the blce fter 5 yers if it is copouded A) Qurterly B) Mothly C) Cotiuously 0) I 986, ucler rector ccidet occurred i Cherobyl i wht ws the the Soviet Uio. The eplosio spred highly toic rdioctive cheicls, such s plutoiu, over hudreds of squre iles, d the goveret evcuted the city d the surroudig re. If 0 pouds of plutoiu where preset i 986, how y pouds will rei i 00 ( yers lter)? How uch will rei fter 00,000 yers? Use is,00 yers. A t,00 = A 0 sice the hlf-life of plutoiu ) Studets prticiptig i psychology eperiet tteded severl lectures o subject d were give e. Every oth for yer fter the e, the studets were retested to see how uch of the teril they reebered. The verge score for the group is give by the hu eory odel f(t) = 75 6 l (t + ), 0 < t < where t is the tie i oths. A) Wht ws the verge score of the origil test? B) Wht ws the verge score t the ed of 6 oths? C) Wht ws the verge score oe yer lter? ) You re to deposit $500 i ccout tht pys 6.75% iterest, copouded cotiuously. How log will it tke your oey to double? ) The uber y of edgered il species i the Uited Sttes fro 990 to 00 c be odeled by y = l t, 0 < t <, where t correspods to the yer 990. Durig which yer did the uber of edgered il species rech 57 [US Fish d Wildlife Service]? ) The uber N of bcteri i culture is odeled by N = 00e kt where t is tie i hours. IF N = 00 whe t = 5, estite the rte k, the use this to fid the out of tie it will tke the bcteri to double i size. 5) Use the Richter scle R log I = to fid the itesity I of erthquke i Cetrl Alsk i 00 where R = 7.9 I 0 d ssue I 0 =. Fid the itesity of erthquke i Hokkido, Jp i 00 with R = 8.. 6) You re drivig o hot dy whe your cr overhets d stops ruig. It overhets t 80 o F d c be drive gi t 0 o F. If r = d it is 80 o F outside, how log will you hve to wit util you c cotiue drivig? 7) You were cookig beef stew. Whe you reoved the stew fro the het, it hd teperture of 00 o F. The roo teperture is 7 o F d the coolig rte of the beef stew is r = How log will it tke to cool the beef stew to servig teperture of 00 o F? 8) Mr. Coley bought his brother i New York ew outdoor theroeter. The theroeter red 75 o F before it ws hug outside. I oe iute, the teperture dropped to 7 o F. If the coolig rte is.7, wht is the true outdoor teperture?