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Chapter 1: Number System
1.3 Indices, Surds and Logarithms
Prepared by: kwkang
Learning Outcomes
(a) Express the rules of indices
(b) Explain the meaning of a surd and its conjugate
(c) Perform algebraic operations on surds
(d) Express the laws of logarithms such as:
(i)
(ii)
(iii)
NMMN aaa logloglog
kwkang@KMK
NMN
Maaa logloglog
MNM a
N
a loglog
(d) Express the laws of logarithms such as:
(i)
(ii)
(iii)
(e) Change the base of logarithm using
Learning Outcomes
NMMN aaa logloglog
NMN
Maaa logloglog
MNM a
N
a loglog
a
MM
b
ba
log
loglog
kwkang@KMK
Indices
aaaaan ...
Bloom: Remembering
kwkang@KMK
n factors
na
index
base
Rules of indices
1.
2.
3.
nmnm aaa
nmnm aaa
nmnm aa )(
Zero index
Bloom: Rememberingkwkang@KMK
Indices
0,10 aa
Negative index
0,1
aa
an
n
Rational index
nn aa
1
mnn
m
aa
Example
Evaluate each of the following without using
calculator.
(a)
(b)
Bloom: Understanding
kwkang@KMK
3
2
27
4
35
125
255
Solution
(a) (b)
Bloom: Understandingkwkang@KMK
32
33
2
327
3
233
23
9
43
325
4
35
5
55
125
255
12
65
5
55
12655
15
5
1
Surds
Bloom: Remembering
kwkang@KMK
Conjugate surds
ba and ba are conjugate surds.
Qbabababa 22
Product of a pair of conjugate surds is a rational number
Bloom: Remembering
kwkang@KMK
Surds
Operations on surds
1.
2.
3.
baab
b
a
b
a
aaaa 2
4.
5.
6.
amnanm
anmanam )(
anmanam )(
Surds
Bloom: Remembering
kwkang@KMK
Rationalising denominators
a
bDivision by surds of the form can be simplified by
multiplying it with (writing 1 as ).
Also known as process of eliminating the surd in the denominator of a fraction.
a
a
a
a
Example
(1) Simplify
(a) (b)
(2) Express in the form of .
(3) Simplify .
(4) Express with a rational denominator.
Bloom: Understanding
kwkang@KMK
20 2213
3475 ba
352331127
13
13
Solution
(1) (a) (2)
(b)
Bloom: Understandingkwkang@KMK
5420
54
52
52
22132213
226
3475
34325
34325
3435
3)45(
39
Solution
(3)
Bloom: Understandingkwkang@KMK
352331127
352331127
353112327
36210
Solution
(4)
Bloom: Understandingkwkang@KMK
13
13
13
13
13
13
2
2
2
13
1323
13
1323
2
324
32
Logarithms
Bloom: Rememberingkwkang@KMK
Index form
xab
Logarithmic form
xba log
Important results
1.
2.
3.
baba
log
01log a
1log aa
5.
6.
110log
1ln e
Logarithms
Bloom: Rememberingkwkang@KMK
Laws of logarithms
1.
2.
3.
NMMN aaa logloglog
NMN
Maaa logloglog
MNM a
N
a loglog
Learning Tip: yx loglog If , then . yx
Example
(1) Express in terms of , and .
(2) Evaluate .
(3) Express as a single logarithm.
(4) Calculate the value of .
Bloom: Understandingkwkang@KMK
2log xyz xlog ylog
zlog
128log2
2log53log yxy aa
30log4
Solution
(1)
(2)
Bloom: Understandingkwkang@KMK
22 loglogloglog zyxxyz
zyx log2loglog 7
22 2log128log
2log7 2
17
7
Solution
(3)
(4)
Bloom: Understandingkwkang@KMK
2log53log yxy aa
25 loglog3log ayxy aaa
5
23log
y
axya
4
23log
y
xaa
453.26021.0
4771.1
4log
30log30log
10
104
Self-check
(1) By using the rules of indices, evaluate
.
(2) Simplify
(a) (b)
(3) Express in the form of .
3835
Bloom: Applyingkwkang@KMK
100
49
636725 ba
4
1
4
13
3273
(4) Simplify .
(5) Express with a rational
denominator.
Self-check
Bloom: Applyingkwkang@KMK
713109761018
225
258
(6) Express in term of , and
.
(7) Evaluate .
(8) If , show that .
(9) Calculate the of without using
calculator.
Self-check
Bloom: Applyingkwkang@KMK
xz
ylog
xlog ylog
zlog
3log2log248log 666
qp 22 log3log2
1 642 pq
3log 81
Answer Self-check
(1)
(2) (a) (b)
(3)
(4)
Bloom: Applyingkwkang@KMK
9
10
7 120
77
719109
(5)
(6)
(7)
(9)
Answer Self-check
Bloom: Applyingkwkang@KMK
3
441017
zxy logloglog2
1
2
4
1
Summary
Surd
Law of Indices
Law of Logarithms
Indices, Surd and
Logarithms
𝑥𝑚 ∙ 𝑥𝑛 = 𝑥𝑚+𝑛
𝑥𝑚 ÷ 𝑥𝑛 = 𝑥𝑚−𝑛
𝑥𝑚 𝑛 = 𝑥𝑚𝑛
𝑥0 = 1
𝑥
𝑦
𝑛
=𝑥𝑛
𝑦𝑛
𝑥−𝑛 =1
𝑥𝑛
𝑥1𝑛 = 𝑛 𝑥
𝑥𝑚𝑛 = 𝑛 𝑥 𝑚
𝑙𝑜𝑔𝑎𝑥𝑦 = 𝑙𝑜𝑔𝑎𝑥 + 𝑙𝑜𝑔𝑎𝑦
𝑙𝑜𝑔𝑎𝑥
𝑦= 𝑙𝑜𝑔𝑎𝑥 − 𝑙𝑜𝑔𝑎𝑦
𝑙𝑜𝑔𝑎𝑥𝑏 = 𝑏𝑙𝑜𝑔𝑎𝑥
𝑙𝑜𝑔𝑎𝑎 = 1𝑎𝑙𝑜𝑔𝑎𝑥 = 𝑥
𝑙𝑜𝑔𝑎𝑥 =𝑙𝑜𝑔𝑏𝑥
𝑙𝑜𝑔𝑏𝑎𝑝 ∙ 𝑞 = 𝑝𝑞
𝑝
𝑞=
𝑝
𝑞
𝑝 𝑟 ± 𝑞 𝑟 = 𝑝 ± 𝑞 𝑟kwkang@KMK Bloom: Remembering
Key Terms
• Indeces
• Zero index
• Negative index
• Rational index
• Surd
• Rationalising denominators
• Conjugate surds
• Logarithms
kwkang@KMK
