Cambridge Additional Mathematics

Answers 471

EXERCISE 5D.1

1algy=xlg 2 blgy=3lgx

clgM=4lgd dlgT=xlg 5

elgy=^12 lgx flgy=lg7+xlg 3

glgS=lg9¡lgt hlgM=2+xlg 7

i lgT=lg5+^12 lgd j lgF=3¡^12 lgn

klgS= lg 200 +tlg 2 l lgy=^12 lg 15¡^12 lgx

2ay=7x b D=2x c F=^5

t

d y=6£ 2 x

eP=px f N=p 31

p

gP=10x^3 h y=^10

x

2

i y=

x^2

10

j T=2k^5 kP=

n^4

9

l y=8£ 16 x

3ay=x

3

2

biy=4 ii y=32

4ay= 100(10

(^13) x
) biy= 100 ii y= 1000
5aIf there is apowerrelationship betweenyandx, for example
y=5x^3 , then there is alinearrelationship between lgy
andlgx.
bIf there is anexponentialrelationship betweenyandx, for
example y=4£ 2 x, then there is alinearrelationship
betweenlgy andx.
EXERCISE 5D.2
1ax=25 b x=67 c x=20 d x=^12564
ex=5 f no solution g x=^98 hno solution
2ax=5 b x=3or 6 c x=2or 4 dx=2
ex=1 f no solution g x=2 hx=4
3ax=8 b x=3 c x=6 d x=4
EXERCISE 5E.1
1a 2 b 3 c^12 d 0 e¡ 1 f^13 g¡ 2
h¡^12
2a 3 b 9 c^15 d^14
3 xdoes not exist such that ex=¡ 2 or 0
4aa ba+1 c a+b dab e a¡b
5ae^1 :^7918 be^4 :^0943 ce^8 :^6995 de¡^0 :^5108
ee¡^5 :^1160 f e^2 :^7081 ge^7 :^3132 he^0 :^4055
i e¡^1 :^8971 j e¡^8 :^8049
6ax¼ 20 : 1 bx=e¼ 2 : 72 c x=1
dx=^1
e
¼ 0 : 368 ex¼ 0 :006 74
fx¼ 2 : 30 gx¼ 8 : 54 h x¼ 0 : 0370
EXERCISE 5E.2
1aln 45 bln 5 cln 4 dln 24
eln 1 = 0 f ln 30 gln(4e) hln
³ 6
e
́
i ln 20 j ln(4e^2 ) kln
³ 20
e^2
́
l ln 1 = 0
2aln 972 bln 200 c ln 1 = 0dln 16 e ln 6
fln
¡ 1
3
¢
gln
¡ 1
2
¢
h ln 2 i ln 16
3 For example, fora, ln 27 = ln 3^3 = 3 ln 3
4 Hint: Ind, ln
μ
e^2
8
¶
=lne^2 ¡ln 2^3
5aD=ex bF=
e^2
p
c P=5e^2 x
dM=e^3 y^2 eB=^14 e^3 t f N=p 31
g
gQ¼ 8 : 66 x^3 hD¼ 0 : 518 n^0 :^4 i T¼
4 : 85
ex
EXERCISE 5F
1ax¼ 3 : 32 bx¼ 2 : 73 c x¼ 3 : 32
dx=4 ex¼ 8 : 00 f x=¡ 5
2ax¼ 1 : 43 bx¼ 1 : 56 c x¼ 3 : 44
dx¼ 5 : 82 ex¼¡ 1 : 34 f x¼ 2 : 37
gx¼ 0 : 275 hx¼ 1 : 81 i x¼ 9 : 64
3ax=ln10 bx= ln 1000 c x=ln0: 15
dx= 2 ln 5 ex=^12 ln 18 f x=0
4ax=^12 ln 300 bx¼ 2 : 85
5ax=¡
lg(0:03)
lg 2
bx=
10 lg
¡ 10
3
¢
lg 5
c x=
¡4lg
¡ 1
8
¢
lg 3
6a 3 : 90 hours b 15 : 5 hours 7bt¼ 6 : 93 hours
8a 50 g b ¼13 200years
9ax=ln2 b x=0 c x=ln2orln 3 dx=0
ex=ln4 fx=ln
³
3+p 5
2
́
orln
³
3 ¡p 5
2
́
10 a(ln 3,3) b(ln 2,5) c (0,2)and(ln 5,¡2)
EXERCISE 5G
1a¼ 2 : 26 b ¼¡ 10 : 3 c¼¡ 2 : 46 d ¼ 5 : 42
2ax¼¡ 4 : 29 bx¼ 3 : 87 c x¼ 0 : 139
3alog 9 26 =^12 log 326 blog 2 11 = 2 log 411
c^6
log 725
= 3 log 57
4ax=^3
p
50 bx=
p
13 c x=49
dx=5 ex=8 f x=16
5b ix=^19 or 9 iix=^12 or 32 iiix=2or 64
EXERCISE 5H
1a iDomain is
fx:x>¡ 1 g,
Range is
fy:y 2 Rg
iii
ii VA i s x=¡ 1 ,
xandy-intercepts 0
iv x=¡^23
v f¡^1 (x)=3x¡ 1
biDomain is
fx:x>¡ 1 g,
Range is
fy:y 2 Rg
iii
ii VA i s x=¡ 1 ,
x-intercept 2 ,
y-intercept 1
iv x=8
v f¡^1 (x)=3^1 ¡x¡ 1
-1-1
y
x
y =log 3 (x + 1)
O
2
11
-1-1
y
x
y = 1 -log 3 (x + 1)
O
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100 100 IB HL OPT
Sets Relations Groups
Y:\HAESE\CAM4037\CamAdd_AN\471CamAdd_AN.cdr Tuesday, 8 April 2014 8:32:53 AM BRIAN