Ch.6 ExponentialandLogarithmicFunctions
6.1 CompositeFunctions
1 FormaCompositeFunction
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Evaluatetheexpressionusingthevaluesgiveninthetable.
1) (f∘g)(4)
x1510 12
f(x) –410 314
x–5–414
g(x) 1–5510
A) 3 B) 10 C) 5 D) Undefined
2) (g∘f)(1)
x1510 12
f(x) –210 315
x–5–213
g(x) 1–8510
A) –8B)5 C)
–2D)10
Evaluatetheexpressionusingthegraphsofy=f(x)andy=g(x).
3) Evaluate(fg)(1).
A) 6 B) –5 C) 3 D) 0
Forthegivenfunctionsfandg,findtherequestedcompositefunctionvalue.
4) f(x)=x+3,g(x)=5x; Find(f∘g)(4).
A) 23 B) 5 7 C) 35 D) 5 35
5) f(x)=2x+4,g(x)=4x2+1; Find(g∘f)(0).
A) 65 B) 12 C) 5 D) 6
Page1
6) f(x)=4x+4,g(x)=2x2+1; Find(f∘g)(2).
A) 40 B) 289 C) 52 D) 163
7) f(x)=18x2–2x ,g(x)=8x–9; Find(f∘g)(1).
A) 20 B) 119 C) 99 D) –16
8) f(x)=5x+8
,
g(x)=–2
/
x; Find(g∘f)(3).
A) –2
23 B) 14
3C) –46
3D) 67
3
9) f(x)=x–6
x,g(x)=x2+9; Find(g∘f)(–2).
A) 25 B) 13 C) 145
16 D) 7
13
10) f(x)=4x+6,g(x)=4x2+3; Find(g∘f)(4).
A) 1939 B) 94 C) 17,959 D) 274
11) f(x)=18x2–8x,g(x)=20x–3; Find(f∘g)(11).
A) 845,866 B) 41,797 C) 804,069 D) 453,530
12) f(t)=t4+24t2+144,g(t)=t+3
3; Find(f∘g)(12).
A) 37 B) 53 C) 1369 D) 780
Forthegivenfunctionsfandg,findtherequestedcompositefunction.
13) f(x)=5x+6
,
g(x)=5x–1; Find(f∘g)(x).
A) 25x+1 B) 25x+11 C) 25x+5 D) 25x+29
14) f(x)=–4x+9
,
g(x)=5x+9; Find(g∘f)(x).
A) –20x+54 B) –20x+45 C) 20x+54 D) –20x–36
15) f(x)=2
x+3,g(x)=1
3x; Find(f∘g)(x).
A) 6x
1+9x B) 1x+3
6x C) 6x
1–9x D) 2x
1+9x
16) f(x)=x–7
2,g(x)=2x+7; Find(g∘f)(x).
A) x B) 2x+7C)x+14 D) x–7
2
17) f(x)=2
x–8,g(x)=5
8x; Find(f∘g)(x).
A) 16x
5–64x B) 5x–40
16x C) 16x
5+64x D) 2x
5–64x
Page2
18) f(x)=x–3
5,g(x)=5x+3; Find(g∘f)(x).
A) x B) 5x+12 C) x+6D)x–3
5
19) f(x)=x+8,g(x)=8x–12; Find(f∘g)(x).
A) 2 2x–1B) 2 2x+1 C) 8 x+8–12 D) 8 x–4
20) f(x)=4x2+6x+5,g(x)=6x–6; Find(g∘f)(x).
A) 24x2+36x+24 B) 24x2+36x+36 C) 4x2+36x+24 D) 4x2+6x–1
21) f(x)=x2–9,g(x)=x2+8; Find(f∘g)(x).
A) x4+16x2+55 B) x4–18x2+89 C) x4+55 D) x4+89
Decidewhetherthecompositefunctions,f∘gandg∘f,areequaltox.
22) f(x)=5x–7,g(x)=x5+7
A) Yes,yes B) No,no C) No,yes D) Yes,no
23) f(x)=x2+3,g(x)=x–3
A) No,no B) No,yes C) Yes,no D) Yes,yes
24) f(x)=x,g(x)=x2
A) Yes,yes B) No,no C) No,yes D) Yes,no
25) f(x)=x–2
4,g(x)=4x+2
A) Yes,yes B) No,no C) Yes,no D) No,yes
26) f(x)=5x,g(x)=x
5
A) Yes,yes B) No,no C) Yes,no D) No,yes
27) f(x)=1
x,g(x)=x
A) No,no B) No,yes C) Yes,no D) Yes,yes
28) f(x)=x+1,g(x)=x2
A) No,no B) No,yes C) Yes,no D) Yes,yes
29) f(x)=x3+8,g(x)=3x–8
A) Yes,yes B) No,no C) No,yes D) Yes,no
Findfunctionsfandgsothatf∘g=H.
30) H(x)=3x+1
A) f(x)=3x;g(x)=x+1 B) f(x)=x+1;g(x)=3x
C) f(x)=3x;g(x)=1 D) f(x)=x;g(x)=x+1
Page3
31) H(x)= 1
x2–3
A) f(x)=1
x,g(x)=x2–3 B) f(x)=x2–3;g(x)=1
x
C) f(x)=1
x2
–9;g(x)=1
xD) f(x)=1
x;g(x)=1
x2
–9
32) H(x)=∣8x+8∣
A) f(x)=∣x∣;g(x)=8x+8 B) f(x)= –∣x∣;g(x)=8x+8
C) f(x)=∣–x∣;g(x)=8x–8 D) f(x)=x;g(x)=8x+8
33) H(x)=(5–2x3)2
A) f(x)=x2;g(x)=5–2x3B) f(x)=5–2x3;g(x)=x2
C) f(x)=(5–2x)3;g(x)=x2D) f(x)=x3;g(x)=(5–2x)2
34) H(x)=1
x2–4
A) f(x)=1
x;g(x)=x2–4 B) f(x)=x2–4;g(x)=1
x
C) f(x)=1
x2;g(x)=–1/4 D) f(x)=1
x2;g(x)=x–4
35) H(x)=10
2x+1
A) f(x)=10
x;g(x)=2x+1 B) f(x)=10
x;g(x)=2x+1
C) f(x)=2x+1;g(x)=10 D) f(x)=10;g(x)=2+1
36) H(x)=|4–3x2|
A) f(x)=|x|;g(x)=4–3x2B) f(x)=4–3x2;g(x)=|x|
C) f(x)=4–3|x|;g(x)=x2D) f(x)=x2;g(x)=4–3|x|
37) H(x)=∣2x+9∣
A) f(x)=∣x∣;g(x)=2x+9 B) f(x)= –∣x∣;g(x)=2x+9
C) f(x)=∣–x∣;g(x)=2x–9 D) f(x)=x;g(x)=2x+9
38) H(x)= 1
x–4
A) f(x)=1
x–4;g(x)=x B) g(x)=x;f(x)=1
x–4
C) f(x)=x–4;g(x)=1
xD) f(x)=1
x–4;g(x)=1
x
Page4
Solvetheproblem.
39) ThepopulationPofapredatormammaldependsuponthenumberxofasmalleranimalthatisits
primaryfoodsource.Thepopulationsofthesmalleranimaldependsupontheamountaofacertainplant
thatisitsprimaryfoodsource.IfP(x)=3x2+5ands(a)=3a+2,whatistherelationshipbetweenthe
predatormammalandtheplantfoodsource?
A) P(s(a))=27a2+36a+17 B) P(s(a))=9a2+12a+9
C) P(s(a))=27a2+18a+17 D) P(s(a))=9a+7
40) AnoilwellofftheGulfCoastisleaking,withtheleakspreadingoiloverthesurfaceofthegulfasacircle.
Atanytimet,inminutes,afterthebeginningoftheleak,theradiusoftheoilslickonthesurfaceis
r(t)=5tft.FindtheareaAoftheoilslickasafunctionoftime.
A) A(r(t))=25πt2B) A(r(t))=5πt2C) A(r(t))=25t2D) A(r(t))=25πt
41) Anairlinecharterservicechargesafareperpersonof$250 plus$20 foreachunsoldseat.Theairplane
holds75passengers.Letxrepresentthenumberofunsoldseatsandwriteanexpressionforthetotal
revenueRforacharterflight.
A) R(x)=(75–x)(250+20x)or18,750+1250x–20x2
B) R(x)=75(250+20x)or18,750+1500x
C) R(x)=(75–x)(250+20x)or18,750+1500x–20x2
D) R(x)=x(250+20x)or250x+20x2
42) ThesurfaceareaofaballoonisgivenbyS(r)=4πr2,whereristheradiusoftheballoon.Iftheradiusis
increasingwithtimet,astheballoonisbeingblownup,accordingtotheformular(t)=4
5t3,t≥0,findthe
surfaceareaSasafunctionofthetimet.
A) S(r(t))=64
25
πt6B) S(r(t))=16
25
πt6C) S(r(t))=64
25
πt3D) S(r(t))=64
25
πt9
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
43) ThesurfaceareaS(insquareinches)ofacylindricalpipewithlength12inchesisgivenbyS(r)=2πr2+24
πr,whereristheradiusofthepiston(ininches).Iftheradiusisincreasingwithtimet(inminutes)
accordingtotheformular(t)=1
6t2,t≥0,findthesurfaceareaSofthepipeasafunctionofthetimet.
44) ThevolumeV(incubicinches)ofacylindricalpipewithlength12inchesisgivenbyV(r)=12πr2,wherer
istheradiusofthepiston(ininches).Iftheradiusisincreasingwithtimet(inminutes)accordingtothe
formular(t)=1
6t2,t≥0,findthevolumeVofthepipeasafunctionofthetimet.
45) Thepricepofacertainproductandthequantitysoldxobeythedemandequationp=–2
3x+200,0≤x≤
300.SupposethatthecostCofproducingxunitsisC=x
20
+800.Assumingthatallitemsproducedare
sold,findthecostCasafunctionofthepricep.
46) Iff(x)=1
2x2+4andg(x)=2x–a,findasothatthegraphoff∘gcrossesthey–axisat36.
Page5
2 FindtheDomainofaCompositeFunction
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findthedomainofthecompositefunctionf∘g.
1) f(x)=8x+56;g(x)=x+10
A) {xxisanyrealnumber} B) {xx≠–17}
C) {xx≠17} D) {xx≠–10
,
x≠–7}
2) f(x)=10
x+6;g(x)=x+10
A) {xx≠–16} B) {xx≠–6}
C) {xx≠–6
,
x≠–10} D) {xxisanyrealnumber}
3) f(x)=x+4;g(x)=5
x+2
A) {xx≠–2} B) {xx≠–6}
C) {xx≠–2
,
x≠–4} D) {xxisanyrealnumber}
4) f(x)=–5
x+8;g(x)=72
x
A) {xx≠0,x≠–9} B) {xx≠0,x≠–8}
C) {xx≠0,x≠–8
,
x≠–9} D) {xxisanyrealnumber}
5) f(x)=6
x;g(x)=–4
x–6
A) {xx≠6}B){xx≠6
,
x≠0}
C) {xx≠0,x≠6
,
x≠1} D) {xxisanyrealnumber}
6) f(x)=x
x+10;g(x)=50
x+4
A) {xx≠–4
,
x≠–9} B) {xx≠–4
,
x≠–10}
C) {xx≠0,x≠–4
,
x≠–9} D) {xxisanyrealnumber}
7) f(x)=x;g(x)=4x+4
A) {xx≥–1} B) {xx≥0}
C) {xx≤–1orx≥0} D) {xxisanyrealnumber}
8) f(x)=4x+4;g(x)=x
A) {xx≥0} B) {xx≥–1}
C) {xx≤–1orx≥0} D) {xxisanyrealnumber}
9) f(x)=x–3;g(x)=3
x–8
A) {x8
<
x≤9} B) {xx≥3
,
x≠8}
C) {xx≠8
,
x≠3} D) {xxisanyrealnumber}
10) f(x)=2
x–7;g(x)=x–2
A) {xx≥2
,
x≠51} B) {xx≥2
,
x≠7}
C) {xx≥2
,
x≠7
,
x≠51} D) {xxisanyrealnumber}
Page6
11) f(x)=2–x;g(x)=2x–1
A) x|–1
2
≤x≤3
2B) allrealnumbers C) {x|x≥2} D) {x|x≤2}
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
Solvetheproblem.
12) Iff(x)=x2andg(x)=–1+5x,find(f∘g)(x)andfindthedomainof(f∘g)(x).
6.2 One–to–OneFunctions;InverseFunctions
1 DetermineWhetheraFunctionIsOne–to–One
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Indicatewhetherthefunctionisone–to–one.
1) {(–18
,
1),(–19
,
–8),(3
,
15)}
A) Yes B) No
2) {(6
,
16),(–19
,
16),(–14
,
9)}
A) Yes B) No
3) {(5
,
4),(6
,
4),(7
,
9),(8
,
5)}
A) Yes B) No
4) {(6
,
–8),(1
,
–7),(–1
,
–6),(–3
,
–5)}
A) Yes B) No
5) {(7
,
–8),(8
,
–7),(–3
,
4),(3
,
–4)}
A) Yes B) No
Usethehorizontallinetesttodeterminewhetherthefunctionisone–to–one.
6)
x
y
x
y
A) Yes B) No
Page7
7)
x
y
x
y
A) Yes B) No
8)
x
y
x
y
A) Yes B) No
9)
x
y
x
y
A) Yes B) No
Page8
10)
x
y
x
y
A) Yes B) No
11)
x
y
x
y
A) Yes B) No
2 DeterminetheInverseofaFunctionDefinedbyaMaporaSetofOrderedPairs
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findtheinverseofthefunctionandstateitsdomainandrange.
1) {(3
,
5),(1
,
6),(–1
,
7),(–3
,
8)}
A) {(5
,
3),(6
,
1),(7
,
–1),(8
,
–3)};D={5
,
6
,
7
,
8 };R={3
,
1
,
–1
,
–3}
B) 3,1
5,1,1
6,–1,1
7,–3,1
8;D={3,1,–1,–3},R=1
5,1
6, 1
7,1
8
C) {(6
,
5),(5
,
–1),(3
,
1),(6
,
7)};D={(6
,
5
,
3};R={(5
,
–1 1
,
7}
D) {(6
,
5),(8
,
–1),(3
,
–1),(6
,
7)};D={6
,
8
,
3};R={5
,
–1
,
7}
2) {(–6
,
9),(–9
,
6),(8
,
–3),(–8
,
3)}
A) {(9
,
–6),(6
,
–9),(–3
,
8),(3
,
–8)}D={9
,
6
,
–3
,
3};R={–6
,
–9
,
8
,
–8}
B) –6,1
9,–9,1
6,8,–1
3,–8,1
3
D={–6,–9,8,–8},R=1
9,1
6,–1
3,1
3
C) {(3
,
8),(8
,
–9),(9
,
–6),(–3
,
–8)};D={3
,
8
,
9
,
–3};R={8
,
–9
,
–6
,
–8}
D) {(3
,
8),(6
,
–9),(9
,
–9),(–3
,
–8)};D={(3
,
6
,
9
,
–3};R={8
,
–9
,
–8}
Page9
3) {(–3,4),(–1,5),(0,2),(2,6),(5,7)}
A) {(4,–3),(5,–1),(2,0),(6,2),(7,5)}D={2,4,5,6,7};R={–3,–1,0,2,5}
B) {(3,4),(1,5),(0,2),(–2,6),(–5,7)};D={3,1,0,–2,–5};R={2,4,5,6,7}
C) {(–3,–4),(–1,–5),(0,–2),(2,–6),(5,–7)};D={–3,–1,0,2,5};R={–7,–6,–5,–4,–2}
D) {(3,–4),(1,–5),(0,–2),(–2,–6),(–5,–7)};D={3,1,0,–2,–5};R={–7,–6,–5,–4,–2}
Findtheinverse.Determinewhethertheinverserepresentsafunction.
4) {(6
,
6),(3
,
7),(1
,
8),(–1
,
9)}
A) {(6
,
6),(7
,
3),(8
,
1),(9
,
–1)};afunction B) {(6
,
6),(7
,
3),(8
,
1),(9
,
–1)};notafunction
C) {(7
,
6),(6
,
1),(6
,
3),(7
,
8)};notafunction D) {(7
,
6),(9
,
1),(6
,
1),(7
,
8)};afunction
3 ObtaintheGraphoftheInverseFunctionfromtheGraphoftheFunction
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Thegraphofaone–to–onefunctionfisgiven.Drawthegraphoftheinversefunctionf–1asadashedlineorcurve.
1) f(x)=3x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page10
2) f(x)=x+5
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
3) f(x)=x3+1
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page11
4) f(x)=2
x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) Functionisitsowninverse
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Usethegraphofthegivenone–to–onefunctiontosketchthegraphoftheinversefunction.Forconvenience,the
graphofy=xisalsogiven.
5)
x
–5–4–3–2–1 12345
y
5
4
3
2
1
-1
-2
-3
-4
-5
(-4, –2)
(-2, 1) (0, 2)
(1, 4)
x
–5–4–3–2–1 12345
y
5
4
3
2
1
-1
-2
-3
-4
-5
(-4, –2)
(-2, 1) (0, 2)
(1, 4)
Page12
A)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
(-2, –4)
(1, –2)
(2, 0)
(4, 1)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
(-2, –4)
(1, –2)
(2, 0)
(4, 1)
B)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
(-4, 2)
(-2, –1) (0, –2)
(1, –4)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
(-4, 2)
(-2, –1) (0, –2)
(1, –4)
C)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
(4, –2)
(2, 1)
(0, 2)
(-1, 4)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
(4, –2)
(2, 1)
(0, 2)
(-1, 4)
D)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
(4, 2)
(2, –1)
(0, –2)
(-1, –4)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
(4, 2)
(2, –1)
(0, –2)
(-1, –4)
4 FindtheInverseofaFunctionDefinedbyanEquation
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Decidewhetherornotthefunctionsareinversesofeachother.
1) f(x)=8x–4,g(x)=x+8
4
A) No B) Yes
2) f(x)=2x+4,g(x)=1
2x–2
A) Yes B) No
3) f(x)=2x–2,g(x)=1
2x+1
A) Yes B) No
4) f(x)=8x2+6,g(x)=x–6
8
A) Yes;Excludetheinterval(–∞
,
6) B) Yes;Novaluesneedtobeexcluded.
C) Yes;Excludetheinterval(–∞
,
8) D) No
Page13
5) f(x)=(x–3)2,x≥3;g(x)=x+3
A) Yes B) No
6) f(x)=(x–6)2,x≥6;g(x)=x+6
A) No B) Yes
7) f(x)=2
x+3,g(x)=3x+2
x
A) No B) Yes C) Yes;Excludethevalue{–3}
8) f(x)=9+x
x,g(x)=9
x–1
A) Yes B) No
9) f(x)=x+6,domain[–6,∞);g(x)=x2+6,domain(–∞,∞)
A) No B) Yes
10) f(x)=x3–4,g(x)=3x+4
A) Yes B) No
Thefunctionfisone–to–one.Finditsinverse.
11) f(x)=8x
A) f–1(x)=x
8B) f–1(x)=8
xC) f–1(x)=–8x D) f–1(x)=8x
12) f(x)=3x+9
A) f–1(x)=x–9
3B) f–1(x)=x+9
3C) f(x)=x–9
3D) f–1(x)=–x+3
9
13) f(x)=8x+4
A) f–1(x)=x–4
8B) f–1(x)=x
8
+4C)f
–1(x)=x+4
8D) f–1(x)=x
8
–4
14) f(x)=4
x
A) f–1(x)=4
xB) f–1(x)=x
4C) f–1(x)=–4x D) f–1(x)=4x
15) f(x)=x2+1,x≥0
A) f–1(x)=x–1,x≥1B)f
–1(x)=x+1,x≥–1
C) f–1(x)=x–1,x≥0D)f–1(x)=x–1,x<0
16) f(x)=6x2+3,x≥0
A) f–1(x)=x–3
6B) f–1(x)=–x–3
6C) f–1(x)=6
x–3D) f–1(x)=6
x–3
17) f(x)=x3+4
A) f–1(x)=3x–4B) f–1(x)=3x+4C) f–1(x)=3x–4D)f
–1(x)=3x+4
Page14
18) f(x)=4x–7
5
A) f–1(x)=5x+7
4B) f–1(x)=5x–7
4C) f–1(x)=5
4x+7D) f–1(x)=5
4x–7
19) f(x)=2
5x–7
A) f–1(x)=2+7x
5x B) f–1(x)=2+7y
5y C) f–1(x)=5x–7
2D) f–1(x)=–7x–2
5x
20) f(x)=6
x+4
A) f–1(x)=–4x+6
xB) f–1(x)=4+6x2
xC) f–1(x)=4+6x
xD) f–1(x)=x
4+6x
21) f(x)=(x–3)3
A) f–1(x)=3x+3B)f
–1(x)=3x–3C)f
–1(x)=x+3D)f
–1(x)=3x+27
22) f(x)=(x+2)3–8.
A) f–1(x)=3x+8–2B)f
–1(x)=3x–2+8
C) f–1(x)=3x+6D) f–1(x)=3x+10
23) f(x)=x+1
A) f–1(x)=x2–1,x≥0B)f
–1(x)=x2+1,x≥0
C) f–1(x)=x–1D) f–1(x)=(x+1)2
24) f(x)=3x–2
A) f–1(x)=x3+2B)f
–1(x)=1
x3+2
C) f–1(x)=x+2D)f
–1(x)=x3+4
25) f(x)=9x+7
–5x+3
A) f–1(x)=–3x+7
–5x–9B) f–1(x)=9x+9
–5x+3C) f–1(x)=–5x–9
–3x+7D) f–1(x)=9x+7
–5x+3
Page15
Findtheinversefunctionoff.Statethedomainandrangeoff.
26) f(x)=3x–2
x+5
A) f–1(x)=5x+2
3–x;domainoff:{x x≠–5};rangeoff:{y y≠3}
B) f–1(x)=x+5
3x–2;domainoff:{x x≠–5};rangeoff:{y y≠2
3}
C) f–1(x)=5x+2
3+x;domainoff:{x x≠–5};rangeoff:{y y≠–3}
D) f–1(x)=3x+2
x–5;domainoff:{x x≠–5};rangeoff:{y y≠5}
Determinei)thedomainofthefunction,ii)therangeofthefunction,iii)thedomainoftheinverse,andiv)the
rangeoftheinverse.
27) f(x)=9x–8
A) f(x):Disallrealnumbers
,
Risallrealnumbers;
f–1(x):Disallrealnumbers,Risallrealnumbers
B) f(x):Disallrealnumbers
,
R={y|y> –8};
f–1(x):Disallrealnumbers,R={y|y<–8}
C) f(x):D={x|x>9}
,
Risallrealnumbers;
f–1(x):D={x|x<9},Risallrealnumbers
D) f(x):D={x|x>9}
,
R={y|y>–8};
f–1(x):D={x|x<9},R={y|y<–8}
28) f(x)=1
x+5
A) f(x):D={x|x≠–5}
,
R={y≠0};
f–1(x):D={x|x≠0},R={y|y≠–5}
B) f(x):Disallrealnumbers
,
Risallrealnumbers;
f–1(x):Disallrealnumbers,Risallrealnumbers
C) f(x):Disallrealnumbers,R=yy≠1
5;
f–1(x):D=xx≠1
5,Risallrealnumbers
D) f(x):D=xx≠1
5,R=yy≠–5 ;
f–1(x):D=xx≠–5 ,R=yy≠1
5
Page16
29) f(x)=3
8x+5
A) f(x):D=xx≠–5
8,R=yy≠0 ;
f–1(x):D=xx≠0 ,R=yy≠–5
8
B) f(x):Disallrealnumbers
,
Risallrealnumbers;
f–1(x):Disallrealnumbers,Risallrealnumbers
C) f(x):D=xx≠3
8,R=yy≠3
5;
f–1(x):D=xx≠3
5,R=yy≠3
8
D) f(x):D=xx≠5
8,R=yy≠–5 ;
f–1(x):D=xx≠–5 ,R=yy≠5
8
30) f(x)=3x–5
A) f(x):D=xx≥5
3,R=yy≥0 ;
f–1(x):D=xx≥0 ,R=yy≥5
3
B) f(x):D=xx≥5
3,Risallrealnumbers;
f–1(x):Disallrealnumbers,R=yy≥5
3
C) f(x):D=xx≥5
3,R=yy≥0 ;
f–1(x):Disallrealnumbers,R=yy≥5
3
D) f(x):D=xx≥0
,
R=yy≥0 ;
f–1(x):D=xx≥0 ,R=yy≥5
3
31) f(x)=3+4x
A) f(x):D=xx≥–3
4,R=yy≥0 ;
f–1(x):D=xx≥0 ,R=yy≥–3
4
B) f(x):D=xx≥–3
4,Risallrealnumbers;
f–1(x):Disallrealnumbers,R=yy≥–3
4
C) f(x):D=xx≥–3
4,R=yy≥0 ;
f–1(x):Disallrealnumbers,R=yy≥–3
4
D) f(x):D=xx≤0
,
R=yy≤0 ;
f–1(x):D=xx≤0 ,R=yy≤–3
4
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
Solvetheproblem.
32) TheprofitPforsellingxitemsisgivenbytheequationP(x)=2x–500.Expressthesalesamountxasa
functionoftheprofitP.
33) Thefunctionf(x)=|x|–5isnotone–to–one.
(a)Findasuitablerestrictiononthedomainoffsothatthenewfunctionthatresultsisone–to–one.
(b)Findtheinverseoff.
Page17
34) TheweightWofabirdʹsbrain(inounces)isrelatedtothevolumeVofthebirdʹsskull(incubicounces)
throughthefunctionW(V)=3.49 3V+1.25.
(a) ExpresstheskullvolumeVasafunctionofbrainweightW.
(b) Predicttheskullvolumeofabirdwhosebrainweighs3oz.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
35) Theaccompanyingtablesrepresentafunctionfthatconvertssecondstohoursandafunctiongthat
convertshourstodays.
x86,400 172,800 259,200 345,600 432,000
f(x) 24 48 72 96 120
x24 48 72 96 120
g(x) 1234 5
Express(f–1∘g–1)(x)symbolically.
A) (f–1∘g–1)(x)=86,400x B) (f–1∘g–1)(x)=x
86,400
C) (f–1∘g–1)(x)=86,400x2D) (f–1∘g–1)(x)=x2
86,400
36) Toremodelabathroom,acontractorcharges$30 perhourplusmaterialcosts,whichamountto$4800.
Therefore,thetotalcosttoremodelthebathroomisgivenbyf(x)=30x+4800wherexisthenumberof
hoursthecontractorworks.Findaformulaforf–1(x).Whatdoesf–1(x)compute?
A) f–1(x)=x
30
–160;Thiscomputesthenumberofhoursworkedifthetotalcostisxdollars.
B) f–1(x)=x
30
–160;Thiscomputesthetotalcostifthecontractorworksxhours.
C) f–1(x)=x
30
–4800;Thiscomputesthenumberofhoursworkedifthetotalcostisxdollars.
D) f–1(x)=x
30
–4800;Thiscomputesthetotalcostifthecontractorworksxhours.
Findaformulafortheinverseofthefunctiondescribedbelow.
37) Asize4dressinCountryCissize24 inCountryD.AfunctionthatconvertsdresssizesinCountryCto
thoseinCountryDisf(x)=x+20.
A) f–1(x)=x–20 B) f–1(x)=x+20 C) f–1(x)=x
–20 D) f–1(x)=x
20
38) Asize6dressinCountryCissize36 inCountryD.AfunctionthatconvertsdresssizesinCountryCto
thoseinCountryDisf(x)=2(x+12).
A) f–1(x)=x
2
–12 B) f–1(x)=x–12
2C) f–1(x)=x
2
+12 D) f–1(x)=x–12
39) Asize60dressinCountryCissize18 inCountryD.AfunctionthatconvertsdresssizesinCountryCto
thoseinCountryDisf(x)=x
2
–12.
A) f–1(x)=2(x+12) B) f–1(x)=2(x–12) C) f–1(x)=2x+12 D) f–1(x)=x+12
Page18
40) 32°Fahrenheit=0°Celsius.AfunctionthatconvertstemperaturesinCelsiustothoseinFahrenheit is
f(x)=9
5x+32.
A) f–1(x)=5
9(x–32) B) f–1(x)=9
5x+32 C) f–1(x)=5
9(x+32) D) f–1(x)=x+32
41) Anorganizationdeterminesthatthecostperpersonofcharteringabusisgivenbytheformula
C(x)=300+6x
x,
wherexisthenumberofpeopleinthegroupandC(x)isindollars.
A) C–1(x)=300
x–6B) C–1(x)=300
x+6C) C–1(x)=300 +x
6D) C–1(x)=6
x–300
6.3 ExponentialFunctions
1 EvaluateExponentialFunctions
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Approximatethevalueusingacalculator.Expressanswerroundedtothreedecimalplaces.
1) 44.6
A) 588.134 B) 447.746 C) 18.400 D) 256.000
2) 5.71.94
A) 29.268 B) 43.699 C) 11.058 D) 20,346.358
3) 5.9542.645
A) 112.039 B) 327.433 C) 15.748 D) 41,040.418
4) 3.2π
A) 38.635 B) 38.983 C) 10.053 D) 36.462
5) 3 7
A) 18.295 B) 7.937 C) 18.520 D) 1093.500
6) e3.78
A) 43.816 B) 37.135 C) 10.275 D) 15.154
7) e–2.4
A) 0.091 B) –6.524 C) –0.091 D) 0.391
Determinewhetherthegivenfunctionisexponentialornot.Ifitisexponential,identifythevalueofthebasea.
8)
x H(x)
–12
08
114
220
326
A) Notexponential B) Exponential;a=2
C) Exponential;a=6 D) Exponential;a=8
Page19
9)
x H(x)
–17
3
01
13
7
29
49
327
343
A) Exponential;a=3
7B) Exponential;a=7
3
C) Exponential;a=3 D) Notexponential
Solvetheproblem.
10) ThefunctionD(h)=7e–0.4hcanbeusedtodeterminethemilligramsDofacertaindruginapatientʹs
bloodstreamhhoursafterthedrughasbeengiven.Howmanymilligrams(totwodecimals)willbe
presentafter11hours?
A) 0.09mg B) 570.16 mg C) 4.33 mg D) 0.67 mg
11) TheformulaP=14.7e–0.21xgivestheaverageatmosphericpressure,P,inpoundspersquareinch,atan
altitudex,inmilesabovesealevel.Findtheaverageatmosphericpressureforanaltitudeof2.3miles.
Roundyouranswertothenearesttenth.
A) 9.1lb/in.2B) 7.8lb/in.2C) 11.0lb/in.2D) 8.4lb/in.2
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
12) Arumorisspreadatanelementaryschoolwith1200studentsaccordingtothemodel
N=1200(1–e–0.16d)whereNisthenumberofstudentswhohaveheardtherumoranddisthenumber
ofdaysthathaveelapsedsincetherumorbegan.Howmanystudentswillhaveheardtherumorafter5
days?
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
13) Thefunctionf(x)=800(0.5)x
/
60modelstheamountinpoundsofaparticularradioactivematerialstoredin
aconcretevault,wherexisthenumberofyearssincethematerialwasputintothevault.Findtheamount
ofradioactivematerialinthevaultafter130years.Roundtothenearestwholenumber.
A) 178pounds B) 581pounds C) 867 pounds D) 185 pounds
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
14) Instrumentsonasatellitemeasuretheamountofpowergeneratedbythesatelliteʹspowersupply.The
timetandthepowerPcanbemodeledbythefunctionP=50e–t/300,wheretisindaysandPisinwatts.
Howmuchpowerwillbeavailableafter378days?Roundtothenearesthundredth.
Page20
15) Acancerpatientundergoingchemotherapyisinjectedwithaparticulardrug.Thefunctio
n
D(h)=4e–0.35hgivesthenumberofmilligramsDofthisdrugthatisinthepatientʹsbloodstreamhhours
afterthedrughasbeenadministered.Howmanymilligramsofthedrugwereinjected?Tothenearest
milligram,howmuchofthedrugwillbepresentafter2hours?
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
16) Agrocerystorenormallysells6jarsofcaviarperweek.UsethePoissonDistributionP(x)=6xe–6
x!
tofind
theprobability(tothreedecimals)ofselling3jarsinaweek.(x!=x·(x–1)·(x–2)·…·(3)(2)(1)).
A) 0.089 B) 0.268 C) 0.178 D) 0.059
17) If7x=3,whatdoes7–3xequal?
A) 1
27 B) 27 C) –27 D) 1
9
18) If4–x=1
3,whatdoes16xequal?
A) 9 B) –9C)3 D)
1
9
2 GraphExponentialFunctions
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Thegraphofanexponentialfunctionisgiven.Matchthegraphtooneofthefollowingfunctions.
1)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) f(x)=4xB) f(x)=4x+1C) f(x)=4x+1 D) f(x)=4x–1
Page21
2)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) f(x)=4x–1B) f(x)=4xC) f(x)=4x–1 D) f(x)=4x+1
3)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) f(x)=2x–1 B) f(x)=2xC) f(x)=2x–1D) f(x)=2x+1
4)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) f(x)=–4xB) f(x)=4xC) f(x)=4–xD) f(x)=–4–x
Page22
5)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) f(x)=3–xB) f(x)=3xC) f(x)=–3xD) f(x)=–3–x
6)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) f(x)=–5–xB) f(x)=5xC) f(x)=–5xD) f(x)=5–x
7)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
A) y=0.32xB) y=2.4xC) y=0.65xD) y=3.5x
Page23
Usetransformationstographthefunction.Determinethedomain,range,andhorizontalasymptoteofthefunction.
8) f(x)=–2x+3+4
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) domainoff:(–∞
,
∞);rangeoff:(–∞
,
4);
horizontalasymptote:y=4
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B) domainoff:(–∞
,
∞);rangeoff:(–∞
,
–4);
horizontalasymptote:y=–4
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C) domainoff:(–∞
,
∞);rangeoff:(–4,∞);
horizontalasymptote:y=4
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D) domainoff:(–∞
,
∞);rangeoff:(–∞
,
–4);
horizontalasymptote:y=–4
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page24
9) f(x)=4(x–4)
x
-6 6
y
6
-6
x
-6 6
y
6
-6
A) domainoff:(–∞
,
∞);rangeoff:(0,∞)
horizontalasymptote:y=0
x
-6 6
y
6
-6
x
-6 6
y
6
-6
B) domainoff:(–∞
,
∞);rangeoff:(0,∞)
horizontalasymptote:y=0
x
-6 6
y
6
-6
x
-6 6
y
6
-6
C) domainoff:(–∞
,
∞);rangeoff:(–∞
,
0)
horizontalasymptote:y=0
x
-6 6
y
6
-6
x
-6 6
y
6
-6
D) domainoff:(–∞
,
∞);rangeoff:(–∞
,
0)
horizontalasymptote:y=0
x
-6 6
y
6
-6
x
-6 6
y
6
-6
Page25
10) f(x)=5–x+2
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
A) domainoff:(–∞
,
∞);rangeoff:(2
,
∞)
horizontalasymptote:y=2
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
B) domainoff:(–∞
,
∞);rangeoff:(5
,
∞)
horizontalasymptote:y=5
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
C) domainoff:(–∞
,
∞);rangeoff:(5
,
∞)
horizontalasymptote:y=5
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
D) domainoff:(–∞
,
∞);rangeoff:(2
,
∞)
horizontalasymptote:y=2
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
Page26
Graphthefunction.
11) f(x)=4x
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
A)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
B)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
C)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
D)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
Page27
12) f(x)=5(x+2)+1.
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
A)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
B)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
C)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
D)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
Page28
13) f(x)=2–x–1
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
A)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
B)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
C)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
D)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
Page29
14) f(x)=1
4
x
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
A)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
B)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
C)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
D)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
Page30
15) f(x)=2
3
x
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
A)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
B)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
C)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
D)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
Page31
16) f(x)= 1
5
·5x.
x
–8–6–4–2 2468
y
8
6
4
2
-2
-4
-6
-8
x
–8–6–4–2 2468
y
8
6
4
2
-2
-4
-6
-8
A)
x
-8 -6 -4 -2 2 4 6 8
y
8
6
4
2
-2
-4
-6
-8
x
-8 -6 -4 -2 2 4 6 8
y
8
6
4
2
-2
-4
-6
-8
B)
x
-8 -6 -4 -2 2 4 6 8
y
8
6
4
2
-2
-4
-6
-8
x
-8 -6 -4 -2 2 4 6 8
y
8
6
4
2
-2
-4
-6
-8
C)
x
-8 -6 -4 -2 2 4 6 8
y
8
6
4
2
-2
-4
-6
-8
x
-8 -6 -4 -2 2 4 6 8
y
8
6
4
2
-2
-4
-6
-8
D)
x
-8 -6 -4 -2 2 4 6 8
y
8
6
4
2
-2
-4
-6
-8
x
-8 -6 -4 -2 2 4 6 8
y
8
6
4
2
-2
-4
-6
-8
Page32
3 DefinetheNumbere
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Graphthefunction.
1) f(x)=–1+ex
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page33
2) f(x)=ex
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
A)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
B)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
C)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
D)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
Page34
3) f(x)=e–x
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
A)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
B)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
C)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
D)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
Page35
4) f(x)=e5x
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
A)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
B)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
C)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
D)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
Page36
5) f(x)=ex–5
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
A)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
B)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
C)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
D)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
Page37
6) f(x)=e3x–1
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
A)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
B)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
C)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
D)
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5
y
5
4
3
2
1
-1
-2
-3
-4
-5
Page38
7) f(x)=3ex
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
A)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
B)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
C)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
D)
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
x
-6 -4 -2 2 4 6
y
6
4
2
-2
-4
-6
8) f(x)=3–e–x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page40
9) f(x)=e–0.3x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 –5 5 10
y
10
5
-5
-10
x
-10 –5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 –5 5 10
y
10
5
-5
-10
x
-10 –5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page41
10) f(x)=5–e–0.79x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 –5 5 10
y
10
5
-5
-10
x
-10 –5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 –5 5 10
y
10
5
-5
-10
x
-10 –5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
4 SolveExponentialEquations
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheequation.
1) 21+2x=32
A) {2} B) {16} C) {4} D) {–2}
2) 22x=1
A) {0} B) {1} C) { 1
22}D)
∅
Page42
3) 4–x=1
16
A) {2} B) {–2} C) 1
4D) 1
2
4) 45–3x=1
256
A) {3} B) 1
64 C) {128} D) {–3}
5) 5x=1
125
A) {–3} B) {3} C) 1
25 D) 1
3
6) 4x=64
A) {3} B) {16} C) {4} D) {2}
7) 2(3x–5)=16
A) {3} B) 1
8C) {8} D) {–3}
8) 1
6
x=1296
A) {–4} B) 1
4C) {4} D) –1
4
9) 9
2
x=4
81
A) {–2} B) 1
2C) {2} D) –1
2
10) 4–x=1
256
A) {4} B) {–4} C) 1
64 D) 1
4
11) 2x2–3=64
A) {3,–3} B) { 35,–35} C) {3} D) {6}
12) 92x·27(3–x)=1
9
A) {–11} B) {–8}
C) 9+87
6,9–87
6D) {10}
Page43
13) 3125x=625
A) 4
5B) 5
4C) 1
5D) 1
4
14) 2(11–2x)=32
A) {3} B) {4} C) {2} D) {–3}
15) 272x–5=94x
A) –15
2B) –2
15 C) 15
2D) 2
15
16) 4x–4=644x
A) –4
11 B) –2
11 C) –1 D) –4
3
17) 125
8
x+1=2
5
x–1
A) –1
2B) 1
2C) –1 D) –1
4
18) 1
3
3x+6=9x–2
A) –2
5B) –4
5C) 10
3D) –1
19) (ex)x·e54=e15x
A) {9
,
6} B) {–9
,
–6} C) {9} D) {6}
20) e4x–1=(e3)–x
A) 1
7B) 1 C) 4
5D) {0}
21) ex–3=1
e6
x+3
A) –15
7B) –21
5C) 6
7D) –6
5
Solvetheproblem.
22) Therabbitpopulationinaforestareagrowsattherateof9%monthly.Ifthereare230rabbitsin
September,findhowmanyrabbits(roundedtothenearestwholenumber)shouldbeexpectedbynext
September.Usey=230(2.7)0.09t
A) 672 B) 671 C) 685 D) 659
Page44
23) Fourbacteriaareplacedinapetridish.Thepopulationwilldouble everyday.Theformulaforthenumber
ofbacteriainthedishondaytis
N(t)=4(2)t
wheretisthenumberofdaysafterthefourbacteriaareplacedinthedish.Howmanybacteriaareinthe
dishfivedaysafterthefourbacteriaareplacedinthedish?
A) 128 B) 40 C) 100 D) 11
24) Thebacteriaina10–litercontainerdoubleevery2 minutes.After57 minutesthecontainerisfull.How
longdidittaketofillaquarterofthecontainer?
A) 53min B) 14.3 min C) 42.8 min D) 28.5 min
25) ThenumberofbooksinasmalllibraryincreasesaccordingtothefunctionB=6600e0.05t,wheretis
measuredinyears.Howmanybookswillthelibraryhaveafter4years?
A) 8061 B) 4613 C) 10,622 D) 10,460
26) Acityisgrowingattherateof0.4%annually.Iftherewere4,492,000 residentsinthecityin1994
,
findhow
many(tothenearestten–thousand)werelivinginthatcityin2000.Usey=4,492,000(2.7)0.004t
A) 4,600,000 B) 290,000 C) 12,130,000 D) 4,630,000
27) Theamountofaradioactivesubstancepresent,ingrams,attimetinmonthsisgivenbytheformula
y=8000(2)–0.3t.Findthenumberofgramspresentin3years.Ifnecessary,roundtothreedecimalplaces.
A) 4.487 B) 4287.094 C) 0.449 D) 428.709
28) Findtheamountinasavingsaccountattheendof6 yearsiftheamountoriginallydepositedis$7000 and
theinterestrateis5.5%compoundedsemiannually.
Use:A=P1+r
n
ntwhere:
A=finalamount
P=$7000(theinitialdeposit)
r=5.5%=0.055(theannualrateofinterest)
n=2(thenumberoftimesinterestiscompoundedeachyear)
t=6(thedurationofthedepositinyears)
A) $9693.49 B) $10,662.84 C) $86,310.00 D) $8237.38
29) Khangborrows$5000atarateof10.5%compoundedmonthly.FindhowmuchKhangowesattheendof
3years.
Use:A=P1+r
n
ntwhere:
A=finalamount
P=$5000(theamountborrowed)
r=10.5%=0.105(theannualrateofinterest)
n=12(thenumberoftimesinterestiscompoundedeachyear)
t=3(thedurationoftheloaninyears)
A) $6841.92 B) $7526.11 C) $181,575.00 D) $5132.40
30) Supposethatf(x)=4x.Whatisf(5)?Whatpointisonthegraphoff?
A) 1024;(5
,
1024) B) 1024;(5
,
4) C) 625;(5
,
625) D) 625;(4
,
625)
Page45
31) Supposethatf(x)=3x.Iff(x)=1
729 ,whatisx?
A) –6 B) 6 C) 3 D) –3
32) Supposethatf(x)=2x+8.Whatisf(3)?Whatpointisonthegraphoff?
A) 16;(3
,
16) B) 8;(3
,
8) C) 8;(8
,
8) D) 16;(3
,
8)
33) Supposethatf(x)=5x+2.Iff(x)=1/627,whatisx?
A) –4 B) 4 C) 2 D) –2
6.4 LogarithmicFunctions
1 ChangeExponentialStatementstoLogarithmicStatements&LogarithmicStatementstoExponentialStatements
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Changetheexponentialexpressiontoanequivalentexpressioninvolvingalogarithm.
1) 52=25
A) log 525=2 B) log 25 5=2 C) log 225 =5 D) log 52=25
2) 4–3=1
64
A) log 41
64
=–3 B) log 1/64 4= –3 C) log –31
64
=4 D) log 4–3=1
64
3) 32=x
A) log 3x=2 B) log x3=2 C) log 2x=3 D) log 32=x
4) 161
/
4=2
A) log 16 2=1
4B) log 216=1
4C) log 42
log 116
=16 D) log 116=1
4
5) 11x=121
A) log 11 121=x B) log x121 =11 C) log 121 11 =x D) log 121 x=11
6) ex=20
A) ln20=x B) log20 x=eC)lnx=20 D) logxe=20
Changethelogarithmicexpressiontoanequivalentexpressioninvolvinganexponent.
7) log 1
/
327=–3
A) 1
3
–3=27 B) 1
3
3=27 C) (–3)1
/
3=27 D) 271
/
3=3
8) log 51
125
=–3
A) 5–3=1
125 B) 35=1
125 C) 5125=3D)
1
125
3=5
Page46
9) log 39=2
A) 32=9B)2
3=9C)3
9=2D)9
2=3
10) log 2x=3
A) 23=xB)3
2=xC)2
x=3D)x
3=2
11) log b81=4
A) b4=81 B) 4b=81 C) 814=bD)81
b=4
12) log 525=x
A) 5x=25 B) x5=25 C) 25x=5D)25
5=x
13) logb49=2
3
A) b2/3=49 B) 492/3=bC)
2
3
b=49 D) b3/2=49
14) lnx=4
A) e4=xB)e
x=4C)4
e=xD)x
4=e
15) ln1
e5
=–5
A) e–5=1
e5B) 1
e5
e=–5C)
1
e5
–5=eD)
–5e=1
e5
2 EvaluateLogarithmicExpressions
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findtheexactvalueofthelogarithmicexpression.
1) log101000
A) 3 B) 1000 C) 30 D) 10
2) log81
64
A) –2 B) 2 C) 8 D) –8
3) log71
343
A) –3 B) 3 C) 49 D) –49
4) log41
64
A) –3B)3 C)
1
3D) –1
3
5) log 81
A) 0 B) 8 C) 1
8D) 1
Page47
6) log1/381
A) –4B)4 C)
1
4D) –1
4
7) log 88
A) 1
2B) 8 C) 1
8D) 1
8) log 10 10
A) 1 B) 10 C) 1
10 D) –1
9) lnl
A) 0 B) 1 C) e D) –1
10) lne
A) 1 B) 0 C) e D) –1
11) lne2
A) 2 B) e C) 1
2D) 1
Useacalculatortoevaluatetheexpression.Roundyouranswertothreedecimalplaces
12) log9
5
A) 0.255 B) 0.588 C) 3.917 D) –0.255
13)
ln7
3
0.26
A) 3.259 B) –2.194 C) –0.629 D) 2.194
14) log2+log2
ln2–ln9
A) –0.400 B) 0.000 C) –0.922 D) 0.208
15) elog90+ln7
log3+ln40
A) 1.742 B) 0.808 C) 0.936 D) –2.260
Solvetheproblem.
16) ThepHofachemicalsolutionisgivenbytheformula
pH=–log10[H+]
where[H+]istheconcentrationofhydrogenionsinmolesperliter.
FindthepHifthe[H+]=5.2×10–9.
A) 8.28 B) 9.72 C) 9.28 D) 8.72
Page48
17) Thelongjumprecord,infeet,ataparticularschoolcanbemodeledbyf(x)=18.3+2.1ln(x+1) wherexis
thenumberofyearssincerecordsbegantobekeptattheschool.Whatistherecordforthelongjump
19yearsafterrecordstartedbeingkept?Roundyouranswertothenearesttenth.
A) 24.6ft B) 24.5 ft C) 24.4 ft D) 20.4 ft
18) Thefunctionf(x)=1+1.5ln(x+1)modelstheaveragenumberoffree–throwsabasketballplayercanmake
consecutivelyduringpracticeasafunctionoftime,wherexisthenumberofconsecutivedaysthe
basketballplayerhaspracticedfortwohours.After13daysofpractice,whatistheaveragenumberof
consecutivefreethrowsthebasketballplayermakes?
A) 5consecutivefreethrows B) 6 consecutivefreethrows
C) 8consecutivefreethrows D) 9 consecutivefreethrows
19) ThenumberofmendyingofAIDS(inthousands)since1987ismodeledbyy=17.3+10.06(lnx), wherex
representsthenumberofyearsafter1987.UsethismodeltopredictthenumberofAIDSdeathsamong
menin1994.Expressanswerroundedtothenearesthundredmen.
A) 36,900men B) 37,000men C) 25,800men D) 26,000men
20) Findasothatthegraphoff(x)=logaxcontainsthepoint(12
,
5).
A) 512 B) 12 5 C) 5
12 D) 12
5
3 DeterminetheDomainofaLogarithmicFunction
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findthedomainofthefunction.
1) f(x)=log(x–5)
A) (5
,
∞)B)(
–5
,
∞) C) (0,∞) D) (1,∞)
2) f(x)=ln(4–x)
A) (–∞
,
4) B) (4
,
∞)C)(
–∞
,
–4) D) (–4
,
∞)
3) f(x)=log3(25–x2)
A) (–5
,
5) B) [–5
,
5] C) (–∞
,
–5)∪(5
,
∞)D)(
–25
,
25)
4) f(x)=log10x+7
x–2
A) (–∞
,
–7)∪(2
,
∞)B)(
–7
,
2) C) (2
,
∞)D)(
–∞
,
–7)
5) f(x)=ln1
x+7
A) (–7
,
∞)B)(7
,
∞) C) (0,∞) D) (1,∞)
6) f(x)=5–ln(9x)
A) (0,∞)B)(
–5
,
9) C) (9
,
∞)D)(
–∞
,
5)∪(9
,
∞)
7) f(x)=lnx
A) (0,∞) B) (1,∞)C)(
–∞
,
1) D) (–∞
,
0)
Page49
4 GraphLogarithmicFunctions
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Thegraphofalogarithmicfunctionisshown.Selectthefunctionwhichmatchesthegraph.
1)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
A) y=logx–3B)y=3–logxC)y=log(3 –x) D) y=log(x–3)
2)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
A) y=log(x–2) B) y=2–logxC)y=log(2 –x) D) y=logx–2
3)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
A) y=log(3–x) B) y=3–logxC)y=log(x–3) D) y=logx–3
Page50
4)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
A) y=3–logxB)y=log(x–3) C) y=log(3 –x) D) y=logx–3
5)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
A) y=–log(–x) B) y=log(–x) C) y= –logxD)y=logx
6)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
A) y=log(–x) B) y=–log(–x) C) y= –logxD)y=logx
Page51
7)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
A) y=–logxB)y=log(–x) C) y= –log(–x) D) y=logx
8)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) y=log 5xB)y=log 5(x+1) C) y=log 5(x–1) D) y=log 5x+1
9)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) y=log 5(x–1) B) y=log 5xC)y=log 5(x+1) D) y=log 5x–1
Page52
10)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) y=log 2x–2B)y=log 2xC)y=log 2(x+2) D) y=log 2(x–2)
11)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) y=log 3(–x) B) y=–log 3xC)y=1–log 3xD)y=log 3x
12)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) y=–log 5xB)y=log 5(–x) C) y=1–log 5xD)y=log 5x
Page53
13)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A) y=1–log 3xB)y=log 3(–x) C) y= – log 3xD)y=log 3x
Page54
GraphthefunctionanditsinverseonthesameCartesianplane.
14) f(x)=log5x
x
-5 5
y
5
-5
x
-5 5
y
5
-5
A)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
B)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
C)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
D)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
Page55
15) f(x)=log1/3x
x
-5 5
y
5
-5
x
-5 5
y
5
-5
A)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
B)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
C)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
D)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
Page56
Graphthefunction.
16) f(x)=–2lnx
x
-5 5
y
5
-5
x
-5 5
y
5
-5
A)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
B)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
C)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
D)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
Page57
17) f(x)=–4–lnx
x
-5 5
y
5
-5
x
-5 5
y
5
-5
A)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
B)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
C)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
D)
x
-5 5
y
5
-5
x
-5 5
y
5
-5
Page58
18) f(x)=2–ln(x+4)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page59
19) f(x)=log 4(x+1)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page60
20) f(x)=–1+log 4x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page61
21) f(x)=2 log 5x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
Page62
22) f(x)=1
2log 3x
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
A)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
B)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
C)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
D)
x
-10 -5 5 10
y
10
5
-5
-10
x
-10 -5 5 10
y
10
5
-5
-10
5 SolveLogarithmicEquations
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheequation.
1) log 5x=3
A) {125} B) {15} C) {8} D) {243}
2) log 28=x
A) {3} B) {16} C) {4} D) {10}
Page63
3) log7x2=4
A) {49
,
–49} B) {2 7,–27} C) {128} D) {2401}
4) log 2(x–3)=3
A) {11} B) {5} C) {12} D) {6}
5) log 3(x–3)=–1
A) 10
3B) –8
3C) 10 D) –8
6) log8(x2–7x)=1
A) {8
,
–1} B) {–8
,
1} C) {8} D) {1}
7) log30(x2–x)=1
A) {–5
,
6} B) {5
,
6} C) {1,30} D) {–5
,
–6}
8) 7ln9x=35
A) e5
9B) e 5
/
9C) {e5}D)
5
ln9
9) 8+8lnx=13
A) {e 5
/
8}B)
e5
8C) ln5
8D) 5
8ln1
10) lnx+9=3
A) {e6–9} B) {e6+9} C) e3
2
+9 D) {e3–9}
11) e5x=2
A) ln2
5B) ln5
2C) 2
5e D) {5ln2}
12) e x+4=6
A) {ln6–4} B) {e6+4} C) {e24} D) {ln10}
TheloudnessL(x),measuredindecibels,ofasoundofintensityx,measuredinwattspersquaremeter,isdefined
asL(x)=10log(x
I0),whereI0=10–12wattpersquaremeteristheleastintensesoundthatahumanearcandetect.
Determinetheloudness,indecibels,ofthesound.
13) AparticularBoeing747jetlinerproducesnoiseataloudnesslevelof113decibels.Findtheintensityleve
l
(roundtothenearesthundredth)inwattpersquaremeterforthisnoise.
A) 0.20wattpersquaremeter B) 0.10wattpersquaremeter
C) 0.40wattpersquaremeter D) 0.80wattpersquaremeter
14) AtarecentPhishrockconcert,soundintensityreachedalevelof0.50wattpersquaremeter.Totheneares
t
wholenumber,calculatetheloudnessofthissoundindecibels.
A) 117decibels B) 123decibels C) 107decibels D) 112decibels
Page64
15) AtarockconcertbyTheWho,themusicregisteredaloudnesslevelof120decibels.Thehumanthreshold
ofpainduetosoundaverages130decibels.Computetheratiooftheintensitiesassociatedwiththesetwo
loudnessleveltodeterminebyhowmuchtheintensityofasoundthatcrossesthehumanthresholdof
painexceedsthatofthisparticularrockconcert.
A) Theintensityofasoundthatcrossesthehumanthresholdofpainis10timesasintenseasthisrock
concert.
B) Theintensityofasoundthatcrossesthehumanthresholdofpainis100timesasintenseasthisrock
concert.
C) Theintensityofasoundthatcrossesthehumanthresholdofpainis1000timesasintenseasthisrock
concert.
D) Theintensityofasoundthatcrossesthehumanthresholdofpainis0.1timesasintenseasthisrock
concert.
16) Youhavetwofriends,JimandAmy.Jimalwaysyellswhenhespeaks,andAmyalwayswhispers.The
loudnessofJimʹsvoiceis120decibels,andtheloudnessofAmyʹsvoiceis20decibels.Determinehow
manytimesasintenseJimʹsvoiceisascomparedatAmyʹs.
A) Jimʹsvoiceis1010timesasintenseasAmyʹs. B) Jimʹsvoiceis100.1timesasintenseasAmyʹs.
C)
J
imʹsvoiceis100timesasintenseasAmyʹs. D)
J
imʹsvoiceis1000timesasintenseasAmyʹs.
TheRichterscaleconvertsseismographicreadingsintonumbersformeasuringthemagnitudeofanearthquake
accordingtothisfunctionM(x)=logx
x0,wherex0=10–3.
17) Whatisthemagnitudeofanearthquakewhoseseismographicreadingis7.8 millimetersatadistanceo
f
100kilometersfromitsepicenter?Roundtheanswertothenearesttenth.
A) 3.9 B) 2.9 C) 2.7 D) 892.1
18) Whatisthemagnitudeofanearthquakewhoseseismographicreadingis7.6millimetersatadistanceo
f
100kilometersfromitsepicenter?Roundtheanswertofourdecimalplaces.
A) 3.8808 B) 0.38808 C) 2.0281 D) 0.20281
19) Whatisthemagnitudeofanearthquakewhoseseismographicreadingis0.94millimetersatadistanceo
f
100kilometersfromitsepicenter?Roundtheanswertofourdecimalplaces.
A) 2.9731 B) 0.9731 C) –0.0269 D) –3.0269
20) Findthemagnitude(toonedecimalplace)ofanearthquakewhoseseismographicreadingis2000
millimetersatadistanceof100kilometersfromitsepicenter.Roundtheanswertothenearesttenth.
A) 6.3 B) 6.4 C) 7.3 D) 5.9
21) Twoearthquakesdifferby0.1whenmeasuredontheRichterscale.Howwouldtheseismographic
readingsdifferatadistanceof100kilometersfromtheepicenter?
A) Theearthquakeofgreatermagnitudehasaseismographicreadingthatis100.1≈1.26timesthatofthe
lesserearthquake.
B) Theearthquakeofgreatermagnitudehasaseismographicreadingthatis100.01≈1.02timesthatof
thelesserearthquake.
C) Theearthquakeofgreatermagnitudehasaseismographicreadingthatis10timesthatofthelesser
earthquake.
D) Theearthquakeofgreatermagnitudehasaseismographicreadingthatis100timesthatofthelesser
earthquake.
Page65
Solvetheproblem.
22) TheformulaD=8e–0.04hcanbeusedtofindthenumberofmilligramsDofacertaindruginapatientʹs
bloodstreamhhoursafterthedrughasbeengiven.Whenthenumberofmilligramsreaches4,thedrugis
tobegivenagain.Whatisthetimebetweeninjections?
A) 17.33hr B) 20.02 hr C) 51.99 hr D) 34.66 hr
23) Between7:00AMand8:00AM,trainsarriveatasubwaystationatarateof5 trainsperhour(0.08 trains
perminute).Thefollowingformulafromstatisticscanbeusedtodeterminetheprobabilitythatatrain
willarrivewithintminutesof7:00AM.
F(t)=1–e–0.08t
Determinehowmanyminutesareneededfortheprobabilitytoreach90%.
A) 28.78min B) 1.14 min C) 8.90 min D) 13.58 min
24) pH=–log10[H+]Findthe[H+]ifthepH=3.4.
A) 3.98×10–4B) 3.98×10–3C) 2.51×10–3D) 2.51×10–4
25) pH=–log10[H+]FindthepHifthe[H+]=3.6×10–3.
A) 2.44 B) 3.56 C) 3.44 D) 2.56
6.5 PropertiesofLogarithms
1 WorkwiththePropertiesofLogarithms
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Usethepropertiesoflogarithmstofindtheexactvalueoftheexpression.Donotuseacalculator.
1) log7712
A) 12 B) 84 C) 7 D) 1
2) lne10
A) 10 B) e C) 10 D) 100
3) lne2
A) 2 B) e C) 2 D) 4
4) log13010+log13013
A) 1 B) 130 C) 10 D) 13
5) log432–log48
A) 1 B) 4 C) 32 D) 8
6) log424–log46
A) 1 B) 4 C) 6 D) 24
7) log612·log12216
A) 3 B) 6 C) 12 D) 216
8) eln14
A) 14 B) e14 C) 13 D) ln14
Page66
9) 10log42–log6
A) 7 B) 42 C) 10,000,000 D) log36
10) eloge264
A) 8 B) 64 C) e8D) e64
Supposethatln2=aandln5=b.Usepropertiesoflogarithmstowriteeachlogarithmintermsofaandb.
11) ln10
A) a+bB)ab C)a–bD)lna+lnb
12) ln5
2
A) b–aB)
a
bC) a+bD)
lna
lnb
13) ln32
A) 5a B) a5C) ab D) 10a
14) ln20
A) 2a+bB)a+bC)4b D)2a+2b
15) ln520
A) 1
5(2a+b) B) 2
5(a+b) C) 2
5(a–b) D) 1
5(a2+b)
2 WriteaLogarithmicExpressionasaSumorDifferenceofLogarithms
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Writeasthesumand/ordifferenceoflogarithms.Expresspowersasfactors.
1) log 13 11
18
A) log 13 11–log 13 18 B) log 13 18 –log 13 11
C) log 13 11+log 13 11 D) log 13 11 ÷log 13 18
2) log 13 6r
s
A) log 13 6+1
2
log 13 r–log 13 s B) log 13 s–log 13 6–1
2
log 13 r
C) log 13 6·1
2
log 13 m÷log 13 s D) log 13 (6 r)–log 13 s
3) log 2x4
y6
A) 4 log 2x–6 log 2y B) 4 log 2x+6 log 2yC)
2
3log 2x
yD) 6 log 2y–4 log 2x
Page67
4) log 4x+2
x8
A) log 4(x+2)–8 log 4x B) log 4(x+2)+8 log 4x
C) 8 log 4x–log 4(x+2) D) log 4(x+2)–log 4x
5) log w13x
5
A) log w13+log wx–log w5 B) log w8x
C) log w13x–log w5 D) log w13 +log wx+log w5
6) log 56x
A) 1
2log 56+1
2log 5x B) log 56+log 5x
C) 1
2log 56x D) log 56+1
2log 5x
7) log 3x
27
A) 1
2log 3x–3B)9–1
2
log 3x C) log 3x–3D)
–3 log 3x
8) ln3ey
A) 1
3
lny+1
3B) y
3C) 1
3
ln3ey+1
3D) 3lny+3
9) log 17
83
s2r
A) 1
8
log 17 3–2log 17 s–log 17 rB)8log 17 3–2log 17 s–log 17 8
C) 1
8
log 17 3–2log 17 s–2log 17 r D) log 17 3–log 17 s–log 17 r
10) log 11 pq
8
A) 1
2log 11 p+1
2log 11 q–1
2log 11 8B)
1
2log 11 p+1
2log 11 q–log 11 8
C) 1
2log 11 pq–1
2log 11 8D)
1
2log 11 p·1
2log 11 q÷1
2log 11 8
11) log 5
8p7q
t2
A) 1
8log 5p+1
7log 5q–2 log 5tB)
1
8log 5p·1
7log 5q÷2 log 5t
C) 8
5log 5p+7
5log 5q–2
5log 5t D) 8 log 5p+7 log 5q–2 log 5t
Page68
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
12) logb3x
5y8
z2
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
13) log1–1
x3
A) logx3–log1–3logx B) log1–3log1–3logx
C) log(x–1)+log(x2+x+1)–3logx D) log(x–1)+log(x2+1)–3logx
14) ln(x+1)(x–8)
(x–4)3
5
/
2,x>8
A) 5
2ln(x+1)+5
2ln(x–8)–15
2ln(x–4) B) 5
2ln(x2+9x–8)–15
2ln(x–4)
C) 5ln(x+1)–2ln(x–8)–15
2ln(x–4) D) ln(x+1)+ln(x–8)+ln5–15ln(x–4)–ln2
15) ln(7x) 11 1+2x
(x–4)9
,x>4
A) ln7+lnx+1
11ln(1+2x)–9ln(x–4) B) ln7+lnx–11ln(1+2x)–9ln(x–4)
C) ln7+lnx+1
11ln(1+2x)–ln9–ln(x–4) D) 7lnx+2
11ln(1+2x)–9ln(x–4)
3 WriteaLogarithmicExpressionasaSingleLogarithm
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Expressasasinglelogarithm.
1) log ct+log cs
A) log cts B) log ct·log cs C) log c(ts) D) log ct
s
2) 5 log bx–log by
A) log bx5
yB) log bx5÷log by C) log b(x5–y) D) log b5x
y
3) 5 log c7+2 log c4
A) log c7542B) log c(35 +8) C) log c75
42D) log c75·log c42
4) ( log aq–log ar)+4 log ap
A) log aqp4
rB) log aqp4r C) log aq
p4rD) log a4qp
r
Page69
5) 2 log bx–3
2log by+1
4log bw–6 log bz
A) log bx2w1
/
4
y3/2z6B) log b2x–3
2y+1
4w–6z
C) log bx2z6
w1/4y3/2 D) log bx2y3
/
2
w1/4z6
6) 3 log 6x+5 log 6(x–6)
A) log 6x(x–6)15 B) log 6x3(x–6)5C) 15log 6x(x–6) D) log 6x(x–6)
7) 3loga(2x+1)–2loga(2x–1)+2
A) logaa2(2x+1)3
(2x–1)2B) loga2(x+1) C) loga(2x+1)+2 D) loga(2x+3)
8) lnx2–2x–35
x–3
–lnx2+2x–15
x+3
+ln(x2–14x+49), x>0
A) ln(x–7)3(x+3)
(x–3)2B) ln3(x–7)(x+3)
2(x–3) C) ln(x–7)3
(x–3)2(x+3)
D) ln3(x–7)
2(x–3)(x+3)
9) 42log66x+log6(42x4)–log642
A) log6x11 B) log6x11
/
6C) log6x10
/
7D) log6x13
/
4
4 EvaluateLogarithmsWhoseBaseIsNeither10Nore
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
UsetheChange–of–BaseFormulaandacalculatortoevaluatethelogarithm.Roundyouranswertothreedecimal
places.
1) log 312.92
A) 2.329 B) 1.111 C) 0.429 D) 4.307
2) log 70.804
A) –0.112 B) –0.095 C) –8.920 D) 8.706
3) log 5.8 271
A) 3.187 B) 2.433 C) 0.314 D) 46.724
4) log 5.2 4.1
A) 0.856 B) 0.613 C) 1.168 D) 0.788
5) log 22
123.9
A) 4.635 B) 2.318 C) 0.452 D) 0.216
UsetheChange–of–BaseFormulaandacalculatortoevaluatethelogarithm.Roundyouranswertotwodecimal
places.
6) log2.9198
A) 4.97 B) 2.30 C) 0.20 D) 68.28
Page70
7) log7.22.3
A) 0.42 B) 0.36 C) 2.37 D) 0.32
8) log 363.4
A) 7.55 B) 3.78 C) 0.24 D) 0.13
9) log325
A) 2.93 B) 0.34 C) 3.22 D) 1.10
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
10) log(2/3)19
Solvetheproblem.
11) Findthevalueoflog34·log45·log56·log67·log78·log89
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
GraphthefunctionusingagraphingutilityandtheChange–of–BaseFormula.
12) logx–3(x+3)
x
123456789
y
5
4
3
2
1
-1
-2
-3
-4
x
123456789
y
5
4
3
2
1
-1
-2
-3
-4
A)
x
123456789
y
5
4
3
2
1
-1
-2
-3
-4
x
123456789
y
5
4
3
2
1
-1
-2
-3
-4
B)
x
123456789
y
5
4
3
2
1
-1
-2
-3
-4
x
123456789
y
5
4
3
2
1
-1
-2
-3
-4
Page71
C)
x
123456789
y
5
4
3
2
1
-1
-2
-3
-4
x
123456789
y
5
4
3
2
1
-1
-2
-3
-4
D)
x
123456789
y
5
4
3
2
1
-1
-2
-3
-4
x
123456789
y
5
4
3
2
1
-1
-2
-3
-4
13) y=log4x
x
2 4 6 8 10 12 14 16 18
y
4
3
2
1
-1
-2
-3
-4
x
2 4 6 8 10 12 14 16 18
y
4
3
2
1
-1
-2
-3
-4
A)
x
2 4 6 8 10 12 14 16 18
y
4
3
2
1
-1
-2
-3
-4
x
2 4 6 8 10 12 14 16 18
y
4
3
2
1
-1
-2
-3
-4
B)
x
2 4 6 8 10 12 14 16 18
y
4
3
2
1
-1
-2
-3
-4
x
2 4 6 8 10 12 14 16 18
y
4
3
2
1
-1
-2
-3
-4
Page72
C)
x
2 4 6 8 10 12 14 16 18
y
4
3
2
1
-1
-2
-3
-4
x
2 4 6 8 10 12 14 16 18
y
4
3
2
1
-1
-2
-3
-4
D)
x
2 4 6 8 10 12 14 16 18
y
4
3
2
1
-1
-2
-3
-4
x
2 4 6 8 10 12 14 16 18
y
4
3
2
1
-1
-2
-3
-4
Page73
14) y=log3(x–2)
x
-8 -6 -4 -2 2 4 6 8
y
8
6
4
2
-2
-4
-6
-8
x
-8 -6 -4 -2 2 4 6 8
y
8
6
4
2
-2
-4
-6
-8
A)
x
-8 -6 -4 -2 2 4 6 8
y
8
6
4
2
-2
-4
-6
-8
x
-8 -6 -4 -2 2 4 6 8
y
8
6
4
2
-2
-4
-6
-8
B)
x
-8 -6 -4 -2 2 4 6 8
y
8
6
4
2
-2
-4
-6
-8
x
-8 -6 -4 -2 2 4 6 8
y
8
6
4
2
-2
-4
-6
-8
C)
x
-8 -6 -4 -2 2 4 6 8
y
8
6
4
2
-2
-4
-6
-8
x
-8 -6 -4 -2 2 4 6 8
y
8
6
4
2
-2
-4
-6
-8
D)
x
-8 -6 -4 -2 2 4 6 8
y
8
6
4
2
-2
-4
-6
-8
x
-8 -6 -4 -2 2 4 6 8
y
8
6
4
2
-2
-4
-6
-8
6.6 LogarithmicandExponentialEquations
1 SolveLogarithmicEquations
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheequation.
1) log 3x=4
A) {81} B) {12} C) {64} D) {1.26}
2) log 5(x+4)=2
A) {21} B) {29} C) {28} D) {36}
Page74
3) log(x+2)=log(5x–3)
A) 5
4B) 5
3C) –5
4D) –1
4
4) log(2+x)–log(x–4)=log3
A) {7} B) 5
2C) {–7} D) ∅
5) log(5x)=log4+log(x–1)
A) –4B) 3
4C) 4 D) –4
9
6) log 3(7x+8)=log 3(7x+2)
A) {0} B) 5
3C) {3} D) ∅
7) log3x+log3(x–24)=4
A) {27} B) {–3,27} C) {53} D) ∅
8) 1
3
log 2(x+6)=log 8(3x)
A) {3} B) {3,0} C) {9} D) ∅
9) log2(3x–2)–log2(x–5)=4
A) {6} B) 38
5C) {18} D) 3
13
10) log 4(x+1)+log 4(x–5)=2
A) {7} B) {7
,
–3} C) {–3} D) {8}
11) log 4(x+1)=2+log 4(x–3)
A) 49
15 B) 4
15 C) –49
15 D) –4
15
12) log 6(x+1)=1–log 6x
A) {2} B) {–2} C) {3} D) {–3}
13) 2+log3(2x+5)–log3x=4
A) 5
7B) 1±46
9C) 1+46
9D) 5
4
Solvetheequation.Expressirrationalanswersinexactformandasadecimalroundedto3decimalplaces.
14) lnx+ln(x+6)=2
A) –6+36+4e2
2
≈1.048 B) –6–36+4e2
2
≈–7.048
C) –6+236+e2
2
≈3.587 D) –6+36+4e2≈2.097
Page75
Solvetheproblem.
15) f(x)=log2(x+1)andg(x)=log2(x–2).
Solvef(x)=8.Whatpointisonthegraphoff?
A) {3},(3
,
8) B) {3},(3
,
10) C) {9},(3
,
8) D) {9},(3
,
4)
16) f(x)=log5(x+1)andg(x)=log5(x–2).
Solveg(x)=2.Whatpointisonthegraphofg?
A) {27},(27
,
2) B) {2},(2
,
27) C) {23},(23
,
2) D) {25},(25
,
2)
17) f(x)=log2(x+3)andg(x)=log2(2x–3).
Solvef(x)=g(x).
A) {6},(6
,
log2(9)) B) {6},(6
,
log2(6)) C) {6},(6
,
log2(3)) D) Nosolution.
18) f(x)=log4(x+2)andg(x)=log4(x–4).
Solvef(x)=g(x).Dothegraphsoffandgintersect?Ifso,where?
A) {6},(6
,
log4(8)) B) {6},(6
,
log4(6))
C) {6},(6
,
log4(2)) D) Nosolution.Nointersection.
19) f(x)=log2(x–5)andg(x)=log2(5x–3).
Solvef(x)+g(x)=6.
A) {7} B) {128} C) {–7} D) {–128}
20) Thefunctionf(x)=1+1.4ln(x+1)modelstheaveragenumberoffree–throwsabasketballplayercan
makeconsecutivelyduringpracticeasafunctionoftime,wherexisthenumberofconsecutivedaysthe
basketballplayerhaspracticedfortwohours.Afterhowmanydaysofpracticecanthebasketballplayer
makeanaverageof7consecutivefreethrows?
A) 72days B) 74days C) 302 days D) 304 days
21) ThepHofasolutionrangesfrom0to14.AnacidhasapHlessthan7.PurewaterisneutralandhasapH
of7.ThepHofasolutionisgivenbypH=–logxwherexrepresentstheconcentrationofthehydrogen
ionsinthesolutioninmolesperliter.FindthehydrogenionconcentrationifthepH=13.
A) 10–13 B) 1013 C) 2.56 D) 0.08
22) ThepHofasolutionrangesfrom0to14.AnacidhasapHlessthan7.PurewaterisneutralandhasapH
of7.ThepHofasolutionisgivenbypH=–logxwherexrepresentstheconcentrationofthehydrogen
ionsinthesolutioninmolesperliter.FindthehydrogenionconcentrationifthepH=7.4.
A) 3.98×10–8B) 3.98×10–7C) 2.51×10–7D) 2.51×10–8
2 SolveExponentialEquations
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheequation.
1) 4x=16
A) {2} B) {4} C) {3} D) {1}
2) 4(1+2x)=64
A) {1} B) {16} C) {4} D) {–1}
Page76
3) 4(7–3x)=1
16
A) {3} B) 1
4C) {4} D) {–3}
4) 3(6+3x)=1
27
A) {–3} B) 1
9C) {9} D) {3}
5) 3·52t–1=75
A) 3
2B) {3} C) 1
2D) 13
10
Solvetheequation.Expressirrationalanswersinexactformandasadecimalroundedto3decimalplaces.
6) 9
5
x=71–x
A) ln7
ln9
5
+ln7
≈0.768 B) ln63
ln35
≈1.165
C) ln9
5
–ln7≈–1.358 D)
ln9
5
+ln7
ln7
≈1.302
Solvetheproblem.
7) f(x)=3x+4andg(x)=3–x+6.
Findthepointofintersectionofthegraphsoffandgbysolvingf(x)=g(x).
A) (1,243) B) (1,81) C) (243
,
1) D) (81
,
1)
8) f(x)=4xandg(x)=14.
Findthepointofintersectionofthegraphsoffandgbysolvingf(x)=g(x).
A) (log414
,
14) B) (log414
,
0) C) (14
,
14) D) (log414
,
4)
9) Findouthowlongittakesa$2500investmenttodoubleifitisinvestedat9%compoundedsemiannually.
Roundtothenearesttenthofayear.UsetheformulaA=P1+r
n
nt.
A) 7.9yr B) 8.1yr C) 7.7 yr D) 8.3yr
10) TheformulaA=294e0.026tmodelsthepopulationofaparticularcity,inthousands,tyearsafter1998.
Whenwillthepopulationofthecityreach423thousand?
A) 2012 B) 2013 C) 2014 D) 2015
11) Thepopulationofaparticularcountrywas26 millionin1981;in1988
,
itwas34million.Theexponential
growthfunctionA=26ektdescribesthepopulationofthiscountrytyearsafter1981.Usethefactthat7
yearsafter1981thepopulationincreasedby8milliontofindktothreedecimalplaces.
A) 0.038 B) 0.297 C) 0.969 D) 0.048
Page77
3 SolveLogarithmicandExponentialEquationsUsingaGraphingUtility
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Useagraphingcalculatortosolvetheequation.Roundyouranswertotwodecimalplaces.
1) log3x+log5x=3
A) {7.09} B) {0.85} C) {1.92} D) {12.82}
2) log4(x+2)–log5(x–1)=1
A) {2.00} B) {1.75} C) {–0.69} D) {2.05}
3) ex=–x
A) {–0.57} B) {0.57} C) {–1.05} D) {1.05}
4) ex–lnx=3
A) {1.14} B) {2.17} C) {1.27} D) {0.57}
5) ex=x5
A) {1.30} B) {–0.79} C) {2.54} D) {–0.71}
6) ex=x2–1
A) {–1.15} B) {0} C) {–0.71} D) {2.54}
7) e2x=x+2
A) {0.45} B) {0.54} C) {2.45} D) {2.54}
8) ln(2x)=–x+2
A) {1.16} B) {1.54} C) {3.16} D) {3.54}
6.7 FinancialModels
1 DeterminetheFutureValueofaLumpSumofMoney
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findtheamountthatresultsfromtheinvestment.
1) $1,000investedat10%compoundedannuallyafteraperiodof10 years
A) $2593.74 B) $1593.74 C) $2357.95 D) $2853.12
2) $1,000investedat11%compoundedsemiannuallyafteraperiodof8 years
A) $2355.26 B) $1355.26 C) $2304.54 D) $2232.48
3) $14,000investedat14%compoundedsemiannuallyafteraperiodof3 years
A) $21,010.22 B) $7010.22 C) $20,741.62 D) $19,635.72
4) $480investedat6%compoundedquarterlyafteraperiodof6 years
A) $686.16 B) $206.16 C) $680.89 D) $676.02
5) $12,000investedat12%compoundedquarterlyafteraperiodof8 years
A) $30,900.99 B) $18,900.99 C) $29,711.56 D) $30,000.96
Page78
Solvetheproblem.
6) Findtheamountowedattheendof8yearsif$5000isloanedatarateof5%compoundedmonthly.
A) $7452.93 B) $12,911.25 C) $9093.60 D) $8060.16
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
7) If$5,000isinvestedfor6yearsat5%,compoundedcontinuously,findthefuturevalue.
8)
J
ohnForgetsalotdeposited$100ata3%annualinterestrateinasavingsaccountfiftyyearsago,andthen
hepromptlyforgothehaddoneit.Recently,hewascleaningouthishomeofficeanddiscoveredthe
forgottenbankbook.Howmuchmoneyisintheaccount?
9) Meikeearned$1565intipswhileworkingasummerjobatacoffeeshop.Shewantstousethismoneyto
takeatriptoEuropenextsummer.Ifsheplacesthemoneyinanaccountwhichpays6.5%compounded
continuously,howmuchmoneywillshehaveinninemonths?
10) Carlahasjustinheritedabuildingthatisworth$250,000.Thebuildingisinahighdemandarea,andthe
valueofthebuildingisprojectedtoincreaseatarateof25%peryearforthenext4years.Howmuchmore
moneywillshemakeifshewaitsfouryearstosellthebuildinginsteadofsellingnow?
2 CalculateEffectiveRatesofReturn
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findtheeffectiverateofinterest.
1) 11%compoundedcontinuously
A) 11.628% B) 11.451% C) 11.089% D) 11.374%
2) 11.7%compoundedcontinuously
A) 12.412% B) 12.151% C) 11.789% D) 12.074%
3) 75.04%compoundeddaily
A) 111.622% B) 75.163% C) 75.245% D) 75.945%
4) 10.25%compoundedmonthly
A) 10.746% B) 10.373% C) 10.455% D) 11.155%
5) 6 1
4%compoundedmonthly
A) 6.29% B) 6.43% C) 6.39% D) 6.25%
6) 7.75%compoundedquarterly
A) 7.978% B) 7.873% C) 7.955% D) 8.655%
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
7) 4 3
4%compoundedquarterly
Page79
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
8) Alocalbankadvertisesthatitpaysinterestonsavingsaccountsattherateof3%compoundedmonthly.
Findtheeffectiverate.Roundanswertotwodecimalplaces.
A) 3.04% B) 3.40% C) 3.44% D) 36%
9) Whichofthetworateswouldyieldthelargeramountin1year:4.7%compoundedquarterly or4.6%
compoundedmonthly?
A) 4.7%compoundedquarterly
B) 4.6%compoundedmonthly
C) Theywillyieldthesameamount.
10) Whichofthetworateswouldyieldthelargeramountin1year:9%compoundedmonthlyor91
4%
compoundedannually?
A) 9%compoundedmonthly
B) 9 1
4%compoundedannually
C) Theywillyieldthesameamount.
3 DeterminethePresentValueofaLumpSumofMoney
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Findthepresentvalue.Roundtothenearestcent.
1) Toget$5600after2yearsat9%compoundedannually
A) $4713.41 B) $886.59 C) $5137.61 D) $8416.8
2) Toget$10,500after9yearsat6%compoundedannually
A) $6214.93 B) $4285.07 C) $6587.83 D) $5918.98
3) Toget$2000after7yearsat9%compoundedsemiannually
A) $1079.95 B) $920.05 C) $1094.07 D) $1128.54
4) Toget$25,000after3yearsat4%compoundedsemiannually
A) $22,199.28 B) $2800.72 C) $22,224.91 D) $22,643.27
5) Toget$6500after10yearsat8%compoundedquarterly
A) $2943.79 B) $3556.21 C) $3010.76 D) $3002.66
6) Toget$10,000after2yearsat12%compoundedmonthly
A) $7875.66 B) $8874.49 C) $11,268.25 D) $5000.00
Solvetheproblem.
7) Whatprincipalinvestedat8%compoundedcontinuouslyfor4yearswillyield$1190?Roundtheanswer
totwodecimalplaces.
A) $1638.78 B) $864.12 C) $1188.62 D) $627.48
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
8) Whatprincipalinvestedat6%,compoundedcontinuouslyfor3years,willyield$1500?Roundtheanswer
totwodecimalplaces.
Page80
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
9) Howmuchmoneyneedstobeinvestednowtoget$2000after4yearsat8%compoundedquarterly?
Expressyouranswertothenearestdollar.
A) $1457 B) $584 C) $1848 D) $2746
4 DeterminetheRateofInterestorTimeRequiredtoDoubleaLumpSumofMoney
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.Roundyouranswertothreedecimals.
1) Whatannualrateofinterestisrequiredtodoubleaninvestmentin12 years?
A) 5.946% B) 2.973% C) 5.776% D) 9.587%
2) Whatannualrateofinterestisrequiredtotripleaninvestmentin9 years?
A) 12.207% B) 6.492% C) 12.983% D) 8.006%
3) Howlongwillittakeforaninvestmenttodoubleinvalueifitearns4.75%compoundedcontinuously?
A) 14.593years B) 15.711 years C) 7.296 years D) 23.129 years
4) Howlongwillittakeforaninvestmenttotripleinvalueifitearns10.5%compoundedcontinuously?
A) 10.463years B) 11.03 years C) 5.231 years D) 6.601 years
Solvetheproblem.
5) Howlongdoesittake$1125totripleifitisinvestedat7%interest,compoundedquarterly?Roundyour
answertothenearesttenth.
A) 15.8years B) 15.8months C) 18.1years D) 18.1months
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
6) Howlongdoesittake$1700todoubleifitisinvestedat5%interest,compoundedmonthly?Roundyour
answertothenearesttenth.
7) Gillianhas$10,000toinvestinamutualfund.Theaverageannualrateofreturnforthepastfiveyearswas
12.25%.Assumingthisrate,determinehowlongitwilltakeforherinvestmenttodouble.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
8) IfEmeryhas$1600toinvestat8%peryearcompoundedmonthly,howlongwillitbebeforehehas
$2900?Ifthecompoundingiscontinuous,howlongwillitbe?(Roundyouranswerstothreedecimal
places.)
A) 7.459yrs,7.434yrs B) 0.644 yrs,0.619 yrs
C) 79.654yrs,7.684yrs D) 0.097 yrs,0.743 yrs
9) Cindywillrequire$12,000in5yearstoreturntocollegetogetanMBAdegree.Howmuchmoneyshould
sheaskherparentsfornowsothat,ifsheinvestsitat11%compoundedcontinuously,shewillhave
enoughforschool?(Roundyouranswertothenearestdollar.)
A) $6923 B) $7121 C) $20,799 D) $3994
10) Traceyboughtadiamondringappraisedat$1200 atanantiquestore.Ifdiamondshaveappreciatedin
valueatanannualrateof12%,whatwasthevalueofthering10yearsago?(Roundyouranswertothe
nearestdollar.)
A) $386 B) $361 C) $3727 D) $109
Page81
11) TheFeldmansboughttheirfirsthousefor$13
,
000. Overtheyearstheymovedthreetimesintobiggerand
biggerhouses.Now,45yearslater,theyarereadytoretireandwantasmallerhouselikethefirstonethey
bought.Ifinflationinpropertyvalueshasaveraged3.6%peryearduringthattime,howmuchwillsucha
housecostthemnow?(Roundyouranswertothenearestdollar.)
A) $63,846 B) $65,690 C) $2647 D) $2573
12) Larryhas$1400toinvestandneeds$1800 in17 years.Whatannualrateofreturnwillheneedtogetin
ordertoaccomplishhisgoal?(Roundyouranswertotwodecimals.)
A) 1.48% B) 1.33% C) 2.33% D) 3.33%
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
13) Aventurecapitalfirminvested$2,000,000inanewcompanyin1995.In1999,theysoldtheirstakeinthe
companyfor$10,500,000.Whatwastheaverageannualrateofreturnontheirinvestment?
14)
J
uliofiguresthathecansave$5000peryear.If,attheendofeachyear,heinveststhemoneyinacertificate
ofdeposit(CD)whichpays7%interestannually,howmuchmoneywillhehavesavedinsixyears?
6.8 ExponentialGrowthandDecayModels;NewtonʹsLaw;LogisticGrowthandDecayModels
1 FindEquationsofPopulationsThatObeytheLawofUninhibitedGrowth
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) ThesizePofasmallherbivorepopulationattimet(inyears)obeysthefunctionP(t)=700e0.19tifthey
haveenoughfoodandthepredatorpopulationstaysconstant.Afterhowmanyyearswillthepopulation
reach2100?
A) 5.78yrs B) 38.13 yrs C) 11.05 yrs D) 13.89 yrs
2) Conservationiststagged130black–nosedrabbitsinanationalforestin1990.In1991
,
theytagged260
black–nosedrabbitsinthesamerange.Iftherabbitpopulationfollowstheexponentiallaw,howmany
rabbitswillbeintherange10yearsfrom1990?
A) 133,120 B) 266,240 C) 139 D) 279
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
3) Therevenueforadot.comcompanyisprojectedtodoubleeachyearforthefirst5years.Iftherevenuefor
thefirstyearis$2million,writeafunctionshowingtherevenueRafterxyears.Whatistherevenuefor
thefourthyear?
4) Inanetworkingmarketingplanforacompany,eachdistributorisexpectedtorecruit3newdistributors.
Jackwasthefirstdistributorhiredbythecompany,soheisconsideredalevel1distributor.The3people
herecruitsareconsideredlevel2distributors.Thepeoplerecruitedbythelevel2distributorsare
consideredlevel3distributors,andsoon.WriteafunctionthatmodelsthenumberofdistributorsDat
eachlevelL.Howmanydistributorswouldtherebeatthefifthlevel?
5) Thebacteriainacontainerquadrupleseveryday.Ifthereareinitially100bacteria,writeanequationtha
t
modelsthenumberofbacteriaAafterddays.Howmanybacteriawilltherebeafter1week?
Page82
6) Theconcentrationofalcoholinapersonʹsbloodismeasurable.SupposethattheriskR(givenasapercent)
ofhavinganaccidentwhiledrivingacarcanbemodeledbytheequation
R=5ekx
wherexisthevariableconcentrationofalcoholinthebloodandkisaconstant.
Supposethataconcentrationofalcoholinthebloodof0.07resultsina10%risk(R=10)ofanaccident.
Findtheconstantkintheequation.
Usingthisvalueofk,whatistheriskiftheconcentrationis0.11?
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
7) During1991,200,000peoplevisitedRaveAmusementPark.During1997,thenumberhadgrownto
834,000.Ifthenumberofvisitorstotheparkobeysthelawofuninhibitedgrowth,findtheexponential
growthfunctionthatmodelsthisdata.
A) f(t)=634,000e0.238t B) f(t)=200,000e0.248t
C) f(t)=634,000e0.248t D) f(t)=200,000e0.238t
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
8) Acultureofbacteriaobeysthelawofuninhibitedgrowth.If140,000bacteriaarepresentinitiallyandthere
are609,000after6hours,howlongwillittakeforthepopulationtoreachonemillion?
9) ThesizePofacertaininsectpopulationattimet(indays)obeysthefunctionP=700e0.03t.Afterhow
manydayswillthepopulationreach1500?
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
10) Thevalueofaparticularinvestmentfollowsapatternofexponentialgrowth.Intheyear2000,you
investedmoneyinamoneymarketaccount.Thevalueofyourinvestmenttyearsafter2000isgivenby
theexponentialgrowthmodelA=7400e0.047t.Howmuchdidyouinitiallyinvestintheaccount?
A) $7400.00 B) $7756.10 C) $347.80 D) $3700.00
11) Thevalueofaparticularinvestmentfollowsapatternofexponentialgrowth.Intheyear2000,you
investedmoneyinamoneymarketaccount.Thevalueofyourinvestmenttyearsafter2000isgivenby
theexponentialgrowthmodelA=3400e0.057t.Whenwilltheaccountbeworth$4271?
A) 2004 B) 2005 C) 2006 D) 2003
12) Thevalueofaparticularinvestmentfollowsapatternofexponentialgrowth.Intheyear2000,you
investedmoneyinamoneymarketaccount.Thevalueofyourinvestmenttyearsafter2000isgivenby
theexponentialgrowthmodelA=6900e0.066t.Bywhatpercentageistheaccountincreasingeachyear?
A) 6.8% B) 7.0% C) 7.2% D) 7.3%
13) Thepopulationofaparticularcountrywas24 millionin1980;in1995
,
itwas36million.Theexponential
growthfunctionA=24ektdescribesthepopulationofthiscountrytyearsafter1980.Usethefactthat15
yearsafter1980thepopulationincreasedby12milliontofindktothreedecimalplaces.
A) 0.027 B) 0.166 C) 0.451 D) 0.037
Page83
2 FindEquationsofPopulationsThatObeytheLawofDecay
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) Thehalf–lifeofsilicon–32is710years.If40 gramsispresentnow,howmuchwillbepresentin1000
years?(Roundyouranswertothreedecimalplaces.)
A) 15.069 B) 36.28 C) 0.002 D) 0
2) Thehalf–lifeofplutonium–234is9hours.If90 milligramsispresentnow,howmuchwillbepresentin6
days?(Roundyouranswertothreedecimalplaces.)
A) 0.001 B) 56.695 C) 0.886 D) 29.689
3) Afossilizedleafcontains36%ofitsnormalamountofcarbon14.Howoldisthefossil(tothenearest
year)?Use5600yearsasthehalf–lifeofcarbon14.
A) 8239 B) 28,899 C) 3599 D) 33,539
4) Thehalf–lifeofaradioactiveelementis130days,butyoursamplewillnotbeusefultoyouafter80%of
theradioactivenucleioriginallypresenthavedisintegrated.Abouthowmanydayscanyouusethe
sample?
A) 312 B) 297 C) 287 D) 302
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
5) Thehalf–lifeofcarbon–14is5700years.Findtheageofasampleinwhich8%oftheradioactivenuclei
originallypresenthavedecayed.
6) Thehalf–lifeofradiumis1690years.If150gramsispresentnow,howlong(tothenearestyear)tillonly
100gramsarepresent?
7) Assumethatthehal
f
–lifeofCarbon–14is5700years.Findtheage(tothenearestyear)ofawoodenaxein
whichtheamountofCarbon–14is30%ofwhatitoriginallyhad.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
8) Strontium90decaysataconstantrateof2.44%peryear.Therefore,theequationfortheamountPof
strontium90aftertyearsisP=P0e–0.0244t.Howlongwillittakefor15gramsofstrontiumtodecayto5
grams?Roundanswerto2decimalplaces.
A) 45.03years. B) 4.50years C) 450.25years D) 40.50years
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
9) Ifasinglepaneofglassobliterates15%ofthelightpassingthroughit,thenthepercentPoflightthat
passesthroughnsuccessivepanescanbeapproximatedbytheequation
P=100e–0.15n
Howmanypanesarenecessarytoblockatleast50%ofthelight?
10) Bob,theincredibleshrinkingman,loseshalfofhisheighteachdayafterhewasexposedtoamysterious
formofcosmicradiation.Howmanydaysbeforeheisliterallyʺknee–hightoagrasshopperʺ?Assumethat
agrasshopperʹskneeis4millimetershighandthatBobis2meterstall.Roundyouranswertothenearest
wholeday.(1000millimeters=1meter)
Page84
11) Theformula
D=8e–0.6h
canbeusedtofindthenumberofmilligramsDofacertaindrugthatisinapatientʹsbloodstreamhhours
afterthedrughasbeenadministered.Thedrugistobeadministeredagainwhentheamountinthe
bloodstreamreaches4milligrams.Whatisthetimebetweeninjections?
12) Between8:30a.m.and9:30a.m.,carsdrivethroughtheCappuccinoExpressatarateof12carsperhour
(0.2perminute).Thefollowingformulafromprobabilitycanbeusedtodeterminetheprobabilitythata
carwillarrivewithintminutesof8:30a.m.
F(t)=1–e–0.2t
Determinehowmanyminutesareneededfortheprobabilitytoreach0.6.
13) Arumorisspreadatanelementaryschoolwith1200studentsaccordingtothemodel
N=1200(1–e–0.16d)whereNisthenumberofstudentswhohaveheardtherumoranddisthenumber
ofdaysthathaveelapsedsincetherumorbegan.Howmanydaysmustelapsefor500tohaveheardthe
rumor?
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
14) ThefunctionA=Aoe–0.0099xmodelstheamountinpoundsofaparticularradioactivematerialstoredin
aconcretevault,wherexisthenumberofyearssincethematerialwasputintothevault.If900poundsof
thematerialareinitiallyputintothevault,howmanypoundswillbeleftafter100years?
A) 334pounds B) 554pounds C) 643 pounds D) 315 pounds
15) ThefunctionA=Aoe–0.0099xmodelstheamountinpoundsofaparticularradioactivematerialstoredin
aconcretevault,wherexisthenumberofyearssincethematerialwasputintothevault.If400pounds
ofthematerialareplacedinthevault,howmuchtimewillneedtopassforonly244poundstoremain?
A) 50years B) 55years C) 60 years D) 100 years
16) Theamountofacertaindruginthebloodstreamismodeledbythefunctiony=y0e–0.40t,wherey0is
theamountofthedruginjected(inmilligrams)andtistheelapsedtime(inhours).Supposethat10
milligramsareinjectedat10:00A.M.Ifasecondinjectionistobeadministeredwhenthereis1milligram
ofthedrugpresentinthebloodstream,approximatelywhenshouldthenextdosebegiven?Expressyour
answertothenearestquarterhour.
A) 3:45P.M B) 12:30P.M C) 5:45P.M D) 5:30P.M
Page85
3 UseNewtonʹsLawofCooling
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) Sandymanagesaceramicsshopandusesa800°F kilntofireceramicgreenware.Afterturningoffherkiln,
shemustwaituntilitstemperaturegaugereaches165°Fbeforeopeningitandremovingtheceramic
pieces.Ifroomtemperatureis70°Fandthegaugereads650°Fin6minutes,howlongmustshewaitbefore
openingthekiln?AssumethekilncoolsaccordingtoNewtonʹsLawofCooling:
U=T+(Uo–T)ekt.
(Roundyouranswertothenearestwholeminute.)
A) 53minutes B) 105minutes C) 77 minutes D) 40minutes
2) Athermometerreading70°Fisplacedinsideacoldstorageroomwithaconstanttemperatureof34°F. If
thethermometerreads64°Fin12minutes,howlongbeforeitreaches60°F?Assumethecoolingfollows
NewtonʹsLawofCooling:
U=T+(Uo–T)ekt.
(Roundyouranswertothenearestwholeminute.)
A) 21minutes B) –7minutes C) 11 minutes D) 14minutes
3) Athermometerreading10°Cisbroughtintoaroomwithaconstanttemperatureof30°C.Ifthe
thermometerreads16°Cafter4minutes,whatwillitreadafterbeingintheroomfor9minutes?Assume
thecoolingfollowsNewtonʹsLawofCooling:
U=T+(Uo–T)ekt.
(Roundyouranswertotwodecimalplaces.)
A) 21.04°C B) 38.96°C C) 2.61°C D) 29.19°C
4) Athermometerreading36°Fisbroughtintoaroomwithaconstanttemperatureof80°F.Ifthe
thermometerreads43°Fafter3minutes,whatwillitreadafterbeingintheroomfor8minutes?Assume
thecoolingfollowsNewtonʹsLawofCooling:
U=T+(Uo–T)ekt.
(Roundyouranswertotwodecimalplaces.)
A) 52.28°F B) 107.72°F C) 28.56°F D) 69°F
5) Acupofcoffeeisheatedto194°andisthenallowedtocoolinaroomwhoseairtemperatureis72°. After
11minutes,thetemperatureofthecupofcoffeeis140°.Findthetimeneededforthecoffeetocooltoa
temperatureof102°.AssumethecoolingfollowsNewtonʹsLawofCooling:
U=T+(Uo–T)ekt.
(Roundyouranswertoonedecimalplace.)
A) 26.4minutes B) 29.7minutes C) 15.1minutes D) 41.1minutes
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
6) Athermometeristakenfromaroomat71°Ftotheoutdoorswherethetemperatureis14°F. Determine
whatthereadingonthethermometerwillbeafter5minutes,ifthereadingdropsto45°F after1minute.
AssumethecoolingfollowsNewtonʹsLawofCooling:
U=T+(Uo–T)ekt.
(Roundyouranswertotwodecimalplaces.)
Page86
7) Thetemperature(indegreesFahrenheit)ofadeadbodythathasbeencoolinginaroomsetat70° is
measuredas88°.Onehourlater,thebodytemperatureis87.5°.Howlong(tothenearesthour)beforethe
firstmeasurementwasthetimeofdeath,assumingthatthebodytemperatureofthedeceasedatthetime
ofdeathwas98.6°.AssumethecoolingfollowsNewtonʹsLawofCooling:
U=T+(Uo–T)ekt.
8) Afullycookedturkeyistakenoutofanovensetat200°C(Celsius)andplacedinasinkofchilledwaterof
temperature4°C.After3minutes,thetemperatureoftheturkeyismeasuredtobe50°C.Howlong(tothe
nearestminute)willittakeforthetemperatureoftheturkeytoreach15°C?Assumethecoolingfollows
NewtonʹsLawofCooling:
U=T+(Uo–T)ekt.
(Roundyouranswertothenearestminute.)
4UseLogisticModels
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) ThelogisticgrowthmodelP(t)=1970
1+38.4e–0.321t
representsthepopulationofabacteriuminaculture
tubeafterthours.Whatwastheinitialamountofbacteriainthepopulation?
A) 50 B) 51 C) 49 D) 55
2) ThelogisticgrowthmodelP(t)=970
1+26.71e–0.348t
representsthepopulationofabacteriuminaculture
tubeafterthours.Whenwilltheamountofbacteriabe600?
A) 10.83hours B) 8.06 hours C) 5.08 hours D) 2.31 hours
3) ThelogisticgrowthmodelP(t)=220
1+21e–0.179t
representsthepopulationofaspeciesintroducedintoa
newterritoryaftertyears.Whenwillthepopulationbe50?
A) 10.17years B) 8.73 years C) –1 years D) –2.44 years
4) ThelogisticgrowthmodelP(t)=380
1+94e–0.195t
representsthepopulationofaspeciesintroducedintoa
newterritoryaftertyears.Whatwillthepopulationbein20years?
A) 131 B) 200 C) 148 D) 380
5) ThelogisticgrowthmodelP(t)=1
1+8.09e–0.868t
representstheproportionofthetotalmarketofanew
productasitpenetratesthemarkettyearsafterintroduction.Whenwilltheproducthave70%ofthe
market?
A) 3.38years B) 2years C) 4.38 years D) 3years
6) In1990,thepopulationofacountrywasestimatedat4million.Foranysubsequentyearthepopulation,
P(t)(inmillions),canbemodeledbytheequationP(t)=240
5+54.99e–0.0208t,wheretisthenumberofyears
since1990.Estimatetheyearwhenthepopulationwillbe21million.
A) approximatelytheyear2088 B) approximatelytheyear2041
C) approximatelytheyear2016 D) approximatelytheyear2093
Page87
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
7) In1992,thepopulationofacountrywasestimatedat5million.Foranysubsequentyear,thepopulation,
P(t)(inmillions),canbemodeledusingtheequationP(t)=250
5+44.99e–0.0208t,wheretisthenumberof
yearssince1992.Determinetheyearwhenthepopulationwillbe39million.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
8) Inatownwhosepopulationis3000,adiseasecreatesanepidemic.Thenumberofpeople,N,infectedt
daysafterthediseasehasbegunisgivenbythefunction
N(t)=3000
1+21.2e–0.54t .Findthenumberofinfectedpeopleafter10days.
A) 2737people B) 142people C) 1000people D) 2000people.
9) Thelogisticgrowthfunctionf(t)=520
1+5.5e–0.18t
describesthepopulationofaspeciesofbutterflies
tmonthsaftertheyareintroducedtoanon–threateninghabitat.Howmanybutterflieswereinitially
introducedtothehabitat?
A) 80butterflies B) 520butterflies C) 6 butterflies D) 2butterflies
10) Thelogisticgrowthfunctionf(t)=800
1+19.0e–0.15t
describesthepopulationofaspeciesofbutterflies
tmonthsaftertheyareintroducedtoanon–threateninghabitat.Whatisthelimitingsizeofthebutterfly
populationthatthehabitatwillsustain?
A) 800butterflies B) 40butterflies C) 19 butterflies D) 1600 butterflies
11) Thelogisticgrowthfunctionf(t)=320
1+5.4e–0.18t
describesthepopulationofaspeciesofbutterflies
tmonthsaftertheyareintroducedtoanon–threateninghabitat.Howmanybutterfliesareexpectedinthe
habitatafter20months?
A) 279butterflies B) 1000 butterflies C) 320 butterflies D) 6400 butterflies
12) Thelogisticgrowthfunctionf(t)=90,000
1+1799.0e–1.4t
modelsthenumberofpeoplewhohavebecomeill
withaparticularinfectiontweeksafteritsinitialoutbreakinaparticularcommunity.Howmanypeople
becameillwiththisinfectionwhentheepidemicbegan?
A) 50people B) 90,000 people C) 1799 people D) 1800 people
13) Thelogisticgrowthfunctionf(t)=16,000
1+399e–1.8t
modelsthenumberofpeoplewhohavebecomeillwitha
particularinfectiontweeksafteritsinitialoutbreakinaparticularcommunity.Howmanypeoplewereill
after9weeks?
A) 15,999people B) 16,000 people C) 360 people D) 16,400 people
14) Thelogisticgrowthfunctionf(t)=70,000
1+2332.3e–1.2t
modelsthenumberofpeoplewhohavebecomeill
withaparticularinfectiontweeksafteritsinitialoutbreakinaparticularcommunity.Whatisthelimiting
sizeofthepopulationthatbecomesill?
A) 70,000people B) 140,000 people C) 2332 people D) 2333 people
Page88
6.9 BuildingExponential,Logarithmic,andLogisticModelsfromDat
a
1 BuildanExponentialModelfromData
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
Solvetheproblem.
1) Thepopulation(inhundredthousands)fortheColonialUnitedStatesinten–yearincrementsfortheyears
1700–1780isgiveninthetable.(Source:1998InformationPleaseAlmanac)
Decade
0
1
2
3
4
Population
251
332
466
629
906
Decade
5
6
7
8
Population
1171
1594
2148
2780
Statewhetherthedatacanbemoreaccuratelymodeledusinganexponentialfunctionoralogarithmic
function.Usingagraphingutility,findamodelforpopulation(inhundredthousands)asafunctionof
decadessince1700.
2) Abiologisthasabacteriasample.Sherecordstheamountofbacteriaeveryweekfor8weeksandfinds
thattheexponentialfunctionofbestfittothedataisA=150·1.79t.Expressthefunctionofbestfitinthe
formA=A0ekt.
3) Amusicstoremanagercollecteddataregardingpriceandquantitydemandedofcassettetapesever
y
weekfor10weeks,andfoundthattheexponentialfunctionofbestfittothedatawasp=25·0.89q.
Expressthefunctionofbestfitintheformp=p0ekq,andusethisexpressiontopredictthequantity
demandedifthepriceis$8.50.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
4) Alifeinsurancecompanyusesthefollowingratetableforannualpremiumsforwomenfortermlife
insurance.Useagraphingutilitytofitanexponentialfunctiontothedata.Predicttheannualpremiumfor
awomanaged70years.
Age 35 40 45 50 55 60 65
Premium $103 $133 $190 $255 $360 $503 $818
A) y=8.94e0.068x,$1044 B) y=6.367e0.068x,$743
C) y=0.0000398x4.06,$1233 D) y= –9306.4+2516.3ln(x),$1723
5) MoneymagazinereportsthatthepercentageoftradingdaysinwhichtheDasdaqlosesorgains2%or
morehasbeenincreasingsince1995indicatingmorevolatilityintheDasdaq.Useagraphingutilitytofit
anexponentialfunctiontothedata.Predictthepercentageoftradingdaysin2001havingsuchswingsin
value.
Year %oftradingdays
1995,02
1996,15
1997,28
1998,318
1999,423
2000,549
A) y=2.353e0.610x,91days B) y=1.278e0.611x,92days
C) y=1.669x1.706,46days D) y=1.849e0.531x,76days
6) Anuclearscientisthasasampleof100mg ofaradioactivematerialwhichhasahalf–lifeinhours.She
monitorstheamountofradioactivematerialoveraperiodofadayandobtainsthefollowingdata.Usea
graphingutilitytofitanexponentialfunctiontothedata.Predicttheamountofmaterialremainingat40
hours.
Hours 0 5 10 15 20 25 30
mg 100 68.3 45.2 31.3 21.5 14.6 9.8.
A) y=100e–0.077x,4.6mg B) y=100e–0.077x,6.7mg
C) y=86e–0.071x,5.0mg D) y=92e–0.0686x,5.9mg
2 BuildaLogarithmicModelfromData
SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.
Solvetheproblem.
1) Datarepresentingthepriceandquantitydemandedforhand–heldelectronicorganizerswereanalyzed
everydayfor15days.Thelogarithmicfunctionofbestfittothedatawasfoundtobep=398–73ln(q).
Usethistopredictthenumberofhand–heldelectronicorganizersthatwouldbedemandediftheprice
were$275.
2) Theratesofdeath(innumberofdeathsper100,000population)for1–4yearoldsintheUnitedStates
between1980–1995aregivenbelow.(Source:NCHSDataWarehouse)
Year RateofDeath
1980 91.4
1985 74.5
1990 69.3
1995 61.3
Alogarithmicequationthatmodelsthisdataisy=822.99–167.55lnxwherexrepresentsthenumberof
yearssince1900.Usethisequationtopredicttherateofdeathfor1–4yearoldsin2005.
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3) Theratesofdeath(innumberofdeathsper100,000population)for20–24yearoldsintheUnitedStates
between1985–1993aregivenbelow.(Source:NCHSDataWarehouse)
Year RateofDeath
1985 134.9
1987 154.7
1989 162.9
1991 174.5
1993 182.2
Alogarithmicequationthatmodelsthisdataisy=57.76+48.56lnxwherexrepresentsthenumberof
yearssince1980andyrepresentstherateofdeathinthatyear.Usethisequationtopredicttheyearin
whichtherateofdeathfor20–24yearoldsfirstexceeds200.
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
4) Afterintroducinganinhibitorintoacultureofluminescentbacteria,ascientistmonitorstheluminosit
y
producedbytheculture.Useagraphingutilitytofitalogarithmicfunctiontothedata.Predictthe
luminosityafter20hours.
Time,hrs 234581015
Luminosity 77.4 60.8 54.5 45.8 30.0 24.3 10.5
A) y=98.75–32.66ln(x),0.91 B) y=112.97–45.97ln(x),–24.74
C) y=107.55–41ln(x),–15.27 D) y=100.5–32.7ln(x),2.54
5) InaPsychologyclass,thestudentsweretestedattheendofthecourseonafinalexam.Thentheywere
retestedwithanequivalenttestatsubsequenttimeintervals.Theiraveragescoresaftertmonthsaregiven
inthetable.
Time,t(inmonths) 1 2345
Score,y(inpercentage) 86.2 85.7 85.4 85.2 85.0
Usingagraphingutility,fitalogarithmicfunctiony=a+blnxtothedata.Usingthefunctionyou
found,estimatehowlongwillittakeforthetestscorestofallbelow84%.Expressyouranswertothe
nearestmonth.
A) 20months B) 10months C) 12months D) 8months
3 BuildaLogisticModelfromData
MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.
Solvetheproblem.
1) Amechanicistestingthecoolingsystemofaboatengine.Hemeasurestheengineʹstemperatureove
r
time.Useagraphingutilitytofitalogisticfunctiontothedata.Whatisthecarryingcapacityofthecooling
system?
time,min 5 10152025
temperature,°F 100 180 270 300 305
A) y=314.79
1+7.86e–0.246x,315°F B) y=314.79
1+7.86e–1.22x ,315°F
C) y=311.63
1+8.1e–0.253x,312°F D) y=306.53
1+7.92e–0.254x,307°F
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Ch.6 ExponentialandLogarithmicFunctions
AnswerKey
6.1 CompositeFunctions
1 FormaCompositeFunction
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2 FindtheDomainofaCompositeFunction
6.2 One–to–OneFunctions;InverseFunctions
1 DetermineWhetheraFunctionIsOne–to–One
2 DeterminetheInverseofaFunctionDefinedbyaMaporaSetofOrderedPairs
3 ObtaintheGraphoftheInverseFunctionfromtheGraphoftheFunction
4 FindtheInverseofaFunctionDefinedbyanEquation
6.3 ExponentialFunctions
1 EvaluateExponentialFunctions
2 GraphExponentialFunctions
3 DefinetheNumbere
4 SolveExponentialEquations
6.4 LogarithmicFunctions
1 ChangeExponentialStatementstoLogarithmicStatements&LogarithmicStatementstoExponentialStatements
2 EvaluateLogarithmicExpressions
3 DeterminetheDomainofaLogarithmicFunction
4 GraphLogarithmicFunctions
5 SolveLogarithmicEquations
6.5 PropertiesofLogarithms
1 WorkwiththePropertiesofLogarithms
2 WriteaLogarithmicExpressionasaSumorDifferenceofLogarithms
3 WriteaLogarithmicExpressionasaSingleLogarithm
4 EvaluateLogarithmsWhoseBaseIsNeither10Nore
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6.6 LogarithmicandExponentialEquations
1 SolveLogarithmicEquations
2 SolveExponentialEquations
3 SolveLogarithmicandExponentialEquationsUsingaGraphingUtility
6.7 FinancialModels
1 DeterminetheFutureValueofaLumpSumofMoney
2 CalculateEffectiveRatesofReturn
3 DeterminethePresentValueofaLumpSumofMoney
4 DeterminetheRateofInterestorTimeRequiredtoDoubleaLumpSumofMoney
6.8 ExponentialGrowthandDecayModels;NewtonʹsLaw;LogisticGrowthandDecayModels
1 FindEquationsofPopulationsThatObeytheLawofUninhibitedGrowth
2 FindEquationsofPopulationsThatObeytheLawofDecay
3 UseNewtonʹsLawofCooling
4 UseLogisticModels
6.9 BuildingExponential,Logarithmic,andLogisticModelsfromDat
a
1 BuildanExponentialModelfromData
2 BuildaLogarithmicModelfromData
3 BuildaLogisticModelfromData
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