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8/12/2019 Chapter5 Indices and Logarithms Notes PDF December 3
2011-4-06 Pm 373k
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SPM Additional Mathematics (3472) Chapter 5 Indices and
Logarithms
http://www.tuitionvalley.com -1-
Chapter 5 Indices and Logarithms1. Indices
(a) xaN = , 00 >> Na ,
(b) 10 =a , aa =1
(c)x
x
aa
1=
(d) nn aa =
1
eg., 331
aa =
(e) nmmnnm aaa == )()(
(f) mnnmn
m
aaa )()()(
11
==
(g) If )__()__( sideHandRightsideHandLeft aa = ,
Then )__()__( sideHandRightsideHandLeft = (Compare the
indices)
Examples,
Given 32 22 =x ,then 532 == xx
(h) nmnm aaa +=
(i) nmnm aaa =
(j) mnnm aa =)(
2. LogarithmxNa =log canbeinterpretedaslogarithmforNofbaseaisx.[
00 >> Na , ]
38log2 = canbeinterpretedaslogarithmfor8ofbase2is3.
Therearetwoimportantlogarithmequations,
01=alog ,[since 01 a= ]
1=aalog [since 1aa = ]
InterchangebetweenindexformandlogarithmformTherearetwomethods:
(a) xNa =log
xaN = (interchange form)
Examples,
38log2 = inlogarithmformcanbewrittenas 328 =
2log10 =x 100102==x
(b) If )__(log)__(log sideHandRightsideHandLeft aa = ,
Then )__()__( sideHandRightsideHandLeft = (Compare the
values)
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8/12/2019 Chapter5 Indices and Logarithms Notes PDF December 3
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SPM Additional Mathematics (3472) Chapter 5 Indices and
Logarithms
http://www.tuitionvalley.com -2-
Examples,
Given ( )3log8log 22 = x ,then 1138 == xx Also,(e) If )__()__(
sideHandRightsideHandLeft > ,
Then )__(log)__(log sideHandRightsideHandLeft aa >
LawsofLogarithmThereare3laws,
(f) nmmn aaa logloglog +=
(g) nm
n
maaa logloglog =
(h) mnm an
a loglog =
Reminder:Toapplylawsoflogarithm, the base must be the
same.Changeofbase
(i)a
bb
c
ca
log
loglog =
Sample questions
NoticethatthebasesareNOTthesame!Applychangeofbase,
VVV
V 22
2
24 log
2
1
2
log
4log
loglog ===
Therefore, 3log2
1log 22 = VT
3loglog 21
22 =
VT [sothatthelawofquotientcanbeapplied]
3log
2
12 =
V
T
Now,changeittoindexform,
823
2
1 ==
V
T VTVT 88 2
1
==
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8/12/2019 Chapter5 Indices and Logarithms Notes PDF December 3
2011-4-06 Pm 373k
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SPM Additional Mathematics (3472) Chapter 5 Indices and
Logarithms
http://www.tuitionvalley.com -3-
Comparethebasesonbothsides,thereisnorelationshipbetween4and7.Hence,applylogbase10onbothsides.
xx 7log4log 101210 =
Uselawofindicestoplacexinlinearform,
7log4log)12( 1010 xx =
7log4log4log2 101010 xx =
4log7log4log2 101010 = xx [gatherthetermswithx]
( ) 4log7log4log2 101010 = x [factorisex]
( )=
=
7log4log2
4log
1010
10x
Comparethebasesonbothsides,32and4arethemultiplesoftwo.Convertthebasesto2,
)68(2)4(5 22 += xx Comparetheindicesonbothsides,
)68(2)4(5 += xx
3
124
121620
121620
=
=
=
+=
x
x
xx
xx
Findtherelationshipbetween4.9with2,7,5andtheirpowers.
52
7
10
499.4
2
==
Therefore,
=
52
7log9.4log
2
55
( )52log7log9.4log 52
55 =
12
)1(2
)5log2(log7log2 555
=
+=
+=
mp
mp
