Cambridge International Mathematics

744 ANSWERS

k^12 l^32 m^13 n^52 o 2

p¡^52 q¡ 2 r 4 s ¡ 2 t¡^52

4alog 4 y=x b log 9 y=x c logay=x

dlogp 3 y=x e log 2 y=x+1 f log 3 y=2n

glog 2 y=¡x h log 3

³y

2

́

=a

5a 2 y=x b 3 y=x c ay=x

dby=n e my=b f 5 T=

a

2

g 32 M=p h b

G

(^5) =m i (
p
b)P=n
6a 7 y=x b x= log 3 y c x= log 0 : 5 y
dx= log 5 z e x=2t f log^2 y
3
=x
gx= 2 log 5 y h x=
3 w
2
i x= log 3 (2z)
j x= log 4 (5y) k x=¡log 2 (10D)
l x=¡1 + log 3 G
7aAsa^0 =1for alla> 0 , thenloga1=0:
bAsa^1 =afor alla> 0 , thenlogaa=1:
EXERCISE 31B
1af¡^1 (x) = log 4 x bf¡^1 (x) = log 10 x
c f¡^1 (x)=¡log 3 x df¡^1 (x) = log 3
³x
2

́

ef¡^1 (x)=7x f f¡^1 (x) = log 5 (2x)

gf¡^1 (x)=2

x

(^3) hf¡^1 (x)=3
x
5
i f¡^1 (x)=(
p
2)x or 2
x
2
2a
bDomain isfxjx 2 Rg Range is fyjy> 0 , y 2 Rg
c Domain isfxjx> 0 ,x 2 Rg Range isfyjy 2 Rg
3 The inverse ofy=ax isx=ay
) y= logax i.e., f¡^1 (x) = logax
4ax¼ 2 : 59 bx¼ 3 : 27 c x¼ 0 : 0401 or 1 : 22
dx¼ 0 : 137 or 1 ex¼ 1 : 32 f x¼ 0 : 0103 or 1
EXERCISE 31C
1alog 316 b log 23 c log 572
dlog 314 e log 312 f log 345
glog 721 h log 4

¡ 36

125

¢

i log 3 (m^2 n^7 )

j log 2

μ

k^5

n^3

¶

2ap+q bq¡p c 2 p d 3 q

ep¡ 2 q fp+2q gp+3¡ 2 q hq+p¡ 2

3ay=u^3 by=

u^3

v

c y=u^2 v^3 dy=2u+v

ey=

2 u

v

fy=

1

u

gy=7v^2 hy=

p

v

u^2

i y=p^363

u

j y=3puv

4a 2 b 2 c ¡ 2 d ¡ 2

EXERCISE 31D.1

1a¼ 100 :^903 b ¼ 101 :^903 c ¼ 102 :^903

d ¼ 10 ¡^0 :^0969 e ¼ 10 ¡^2 :^0969 f ¼ 10 ¡^0 :^523

g¼ 10 ¡^1 :^523 h ¼ 10 ¡^4 :^523 i ¼ 101 :^699

j ¼ 10 ¡^3 :^301

2alog 30 b log 5 c log 12 d log(^54 ) elog 1 = 0

f log 30 g log 4 h¡log 6 or log(^16 ) i log( 5121 )

j log 2000 klog( 12564 ) l log 20 mlog 5 n log 20

o log(14 000)

3 log 30 = log(3£10) = log 3 + log 10, etc.

4a 3 b 2 c^23 d ¡ 1 e¡ 1 f ¡ (^112)
g
b
3 h
2
a
6 Hint: 2400 = 2^3 £ 3 £ 52
EXERCISE 31D.2
1alogy= loga+ 2 logb b logy= 2 loga¡logb
c logy= logd+^12 logp d logM= 2 loga+ 5 logb
e logP=^12 (loga+ logb) f logQ=^12 logm¡logn
glogR= loga+ logb+ 2 logc
h logT= log 5 +^12 (logd¡logc)
i logM= loga+ 3 logb¡^12 logc
2aQ=10x+2 or Q= 100(10x)
b J=10^2 x¡^1 or J= 101 (100x)
c M=10^2 ¡x or M=
100
10 x
d P¼ 100 :301+x or P¼2(10x)
e R¼ 10 x+1:^477 or R¼30(10x)
f K¼ 10
(^12) x+1
or K¼10(10
x 2
)
3aM=ab bN=d
e
c F=x^2 dT=pp
e D=
1
g
fS=
1
b^2
gA=
B
C^2
hp^2 q=s
i m
3
d
=n
p^2
j pm
n
=p^2 kN=10t l P=^100
x
EXERCISE 31E
1ax¼ 1 : 903 b x¼ 3 : 903 c x¼¡ 1 : 602
d x¼ 2 : 659 e x¼¡ 0 :057 30 f x¼¡ 3 : 747
2ax=1: 585 b x=3: 322 c x=8: 644
d x=¡ 7 : 059 e x=4: 292 f x=¡ 0 :099 97
gx=6: 511 h x=4: 923 i x=49: 60
j x=4: 376 k x=8: 497 l x= 230: 7
3a 17 : 0 hours b 64 : 6 hours
4a 17 : 6 hours b 36 : 9 hours
5a 4 : 59 years¼ 4 years 7 months
b 9 : 23 years¼ 9 years 3 months
6 ¼ 1 : 59
7a¼ 3 : 58 b¼ 4 : 19 c ¼ 2 : 02 d¼¡ 3 : 99
8ax=4 bx= 251 c x=2 dx= 101
y yx¡=¡
x
-2-1 1 2
2
1
-1
-2
O
yx¡=¡logc
y¡=¡3x
IB MYP_3 ANS
cyan magenta yellow black
(^05255075950525507595)
100 100
(^05255075950525507595)
100 100
Y:\HAESE\IGCSE01\IG01_an\744IB_IGC1_an.CDR Thursday, 20 November 2008 2:06:22 PM PETER