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differ s from th at of its surroundin gs will either cool downor heat up to th at of it s surroundin gs. Supp ose a bodythat h as a temp erature T I is pl aced in surround ings withtemperature To' The b ody will either coo l or w arm to tem-peratur e TCI) after tim e t , in minutes , where

TCt) = To + ITI - Tole-kt .

A cup of coff ee whos e temper ature is 105 F is plac ed ina freeze r whose temperature i s 3YF. After 5 minut es, itstemperature i s 70e . What will its t emperatur e be after10 minut es?

5UMMARY AHD REV IEW

CHAPTER 5 SUMMARY AND REVIEW 349

CHALLENGE

39. When was dIe murder committed? The poli ce discoverthe bod y of a math prof essor. Criti cal to solving the crim eis determining when th e murd er was committ ed. Th epolice c a1\ the coroner , who ar rives at 12: 00 P.M. Thecoron er immedi ately takes the temper ature of th e bod yand find s it to b e 94.6. The cor oner takes the temper a-ture 1 hour later and find s it to be 93.4'. The temperatur eof the r oom is 70e . When was th e murd er committ ed?(Use Newton's L aw of Cooling, Exercis e 38.)

Inver se relation , p. 284One-to-one fun ction , p. 286Inverse functi on, p. 286Hori zontal-lin e test, p. 287

Expon ential function , p. 29 5Base, p. 296Loga rithmi c [uncti on, p. 303Comm on logarithm , p. 317

Natur al logarithm , p. 319Exp onential equati on, p. 327Logarithmi c equati on, p. 329

1. Find th e inver se o f the relation H given by

H = {( -4,5), (2, -3),0,7), (8,8), (5, -4)}.

Writ e an equation of the inverse .

2. y = 3x 2 + 2x - 1 3. Y = F+- i4. Which of the f ollowin g have in verses th at are fun ction s?

a) b)y

x

c) d)

x

y

Find a formul a for F 1 (x ).

J ~5. j(x) = 2 2

7. Findf CF "(n)):

j(x) = 3 + 2.

6. j(x ) = x3

+ 88. Find h -I(h (t)):

h ex) = x ' 7 + X6 5

Graph .

(I)". Y = og2 (x - 1 ) 10 . Y = '211. j(x) = 3(1 - e - X), for n onnegativ e valu es of x

12. j(x) = n (x - 4)

13. F ind lo g , 10 usin g common logarithm s.

14. Find lo g6 2 using natural l ogarithm s.

15. Write an exponential equ ation equivalen t to logs = l16. Write a l oga rithmi c equ ation equ ivalent to 7 2.3 = x.

17. W rite an equi valent expression containin g a sin gle log-arithm:

x

1 3210gh a + 210gb C - 4 log, d .

18 . Express in t erms of log arithms of M and N:

logfol/N.

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350 CHAPTER 5 EXPON ENTIAL AN D LOGARITHMI C FUNCTIONS

Given that loga 2 = 0.301 , lo ga 3 = 0.477, and log . 7 = 0.845,find eac h of th e fo llo w in g .

19. log. 18

21. l og, t20 . loga 122. loga fi

S impli fy .

23. log l2 12 x'+1

Solve.

25. log x 64 = 326. log l64 = x

27. logs 12 5 = x

28 .3 1 -x= 9 lx

29. eX = 80

30 . log Xl = og x

31. log (x 2 - 1 ) - log (x - 1 ) = 132. log 2 + 210gx = og (5x + 3)33. logl (x - 1) + log , (x + 1) = 3

24. logs 8.. /9

34. How m any ye ars w ill it tak e an i nvestm ent of $1000 todo uble if int erest i s co mpound ed annuall y at 13%7

35. What i s th e loudn ess, in de cib els, o f a so und wh ose in-tens it y is 100 0I o?

36. Th e half -lif e o f a r adi oac ti ve s ub stan ce is 15 da ys . H owmu ch of a 25-g ram sampl e w ill r emain r adi oacti ve a ft er30 days?

37. Forgetting. In an art class , st udents w ere tested at theend of th e co urse on a final e xam . Theyw ere tested a ga inaft er 6 m onths. Th e fo rgettin g formul a was determ inedto be

Set) = 82 - 38 log (t + 1),w here t is t he tim e, in m onth s, aft er t ak in g th e fir st tes t.a) Wh at was th e ave rag e s co re when th ey initi a ll y too k

the t es t, t = O?b) What wa s the averag e scor e a fter 6 m onths?c) A ft er what tim e wa s the average sc ore 5 4?

38 . The cost of a prime-rib dinner . Th e average cost C o fa p rim e-r ib dinn er was $ 4 .65 in 1962 . In 1986 , it w as$15.8 1. Ass ume th at t he gr ow th f o ll owe d the ex ponentialg row th fun ction.

a) Find k and writ e the exp onential gr owth functi on.b) How mu ch will a p rime-rib d inner co st in 2010 ?c) Wh en w ill th e ave rage cos t of a pr im e-rib dinn er be

$20?

d) W hat i s t he doubling tim e?

39. The population of a cit y doubled in 18 year s. What wa sth e ex pon enti a l growth r ate?

40. H ow long w ill it t ake $76 00 to d oubl e it self if it i s i n-ves ted at 8 .6%, co mp ound ed c ont inu ousl y?

41. How o ld i s a s keleton th at ha s lo st 2 7% of it s ca rb on-1 4?

42. Wh at is th e pH o f a sub stance wh ose hydr oge n ion c on-centration is 3 .8 x 10 -7 moles p er liter?

43. An ea rthquak e has an int ensit y of lO s1 0 ' Wh at is i ts m ag -nit ude on th e Ri chter sca le?

Find eac h of th e fo ll ow in g com mon and natur al l oga rithm susing a ca lcul ato r.

44. lo g 0.002] 6

46. In87 ,380

45. log 1 ,3 42 ,000

47. In 0.00002776

SYNTHESIS

Solve .48. Il og 4x] = 3

Graph.

50. y = Il og3 x]

49. log x = nx

51. Y = le X- 41

Find th e domain .

1 853. j(x) = e 4 x _ 102. j(x ) = f,

v51n x-6

THINKING AND WRIT ING

1. Supp ose you we re try in g to c on vin ce a fello ws tud ent th atlog2(x + 3) = f. log2 X + log, 3. G ive as m any ex plan a-ti ons a s you ca n.

2. D escribe the diff erenc e between r 1 ex) and [ j(x) ]- I.3 . D esc rib e the diff erenc e betwe en j(x) = ]X and g(x) = 3 .4. L oo k up d ata for as m any pr ece din g yea rs as yo u ca n

rega rdin g th e numb er of c ase s of th e di sease A IDS th atar e occ urrin g a nnu all y . H ow mi ght yo u kn ow w hethe r a nexponenti a l function fit s the dat a? Ifso , use so m e of th edata to find a function that fits and predi ct th e numb erof AIDS cas es in the yea rs 2000 and 2010 . Then u seoth er part s of th e da ta to find an oth er fun cti on , mak epr edi ction s for th e numb er of A ID S c as es in 2000 and

2010 , and co mpare .