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Mathematics DM015
Topic 1 : Number System
______________________________________________________________________________
LECTURE 1 OF 7
TOPIC
: 1.0 NUMBER SYSTEM
SUBTOPIC
: 1.1 Real Numbers
LEARNING
OUTCOMES
: At the end of the lesson, students should be able to
(a)
define and understand natural numbers, whole numbers, integers,
prime numbers, rational numbers and irrational numbers.
(b)
represent rational and irrational numbers in decimal form,
(c)
represent the relationship of number sets in a real numbers
diagrammatically showing N  W  Z  Q and Q  Q = ,
where N, W , Z , Q , Q and R represent the set of natural, whole,
integer, rational and real numbers respectively.
SET INDUCTION:
1.1
Real Numbers
The lecturer gives the situation such as;
i.
how many children in the family,
ii.
what their heights,
iii.
“Assume that a woman wants to buy fishes. The total price of the fishes is RM10, but
the woman has only RM5. So she owes RM5 from the seller. (RM5 – RM10 = RM5).
In conclusion, numbers can be classified into different types of numbers where dates of birth
belong to natural numbers, the height is rational numbers and –RM5 is a negative number.
CONTENT
A)
Bil.
TYPES OF NUMBERS
Type of numbers
1
Natural numbers
2
Prime numbers
3
Whole numbers
Symbol
N
W






Characteristics
Example
no zero
N = {1, 2, 3 …}
no negative numbers
no decimal numbers
Greater than 1 &;
Only can be divided by itself = {2, 3, 5, 7 …}
and 1
W = {0, 1, 2, 3 …}.
‘natural numbers’ + zero
1
Mathematics DM015
Topic 1 : Number System
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 Negatives numbers
Z = {…, -3, -2, -1, 0, 1,
4
Integers
Z
 Has zero
2, 3 …}
 No decimal numbers
 Positive whole numbers
5
 Positive
Z+
Z + = {1, 2, 3 …}
 Without zero
 Negative whole numbers
Z   ..., 3, 2, 1
6
 Negative
Z
 Without zero
 Numbers can be multiply or = {2k, k  Z}
7
 Even
divide by 2.
= {2, 4, 6, 8, …}
= {2k + 1, k  Z}
8
 Odd
 Other than even numbers
= {1, 3, 5, 7, …}
a

 Can be written in quotient
Q=  ; a, b  Z , b  0  .
b

@ ratio of two integers.
9
Rational numbers
Q
 For terminating or repeating
3 5 4 
=  , ,  ,...
decimals.
2 1 9 
 Numbers are not repeating
or terminating.
10 Irrational numbers
Q
= 3, 10,  , e
 Cannot be expressed as
quotient /ratio.
 The real number consists of
11 The real number
rational numbers and
 Q Q
irrational numbers.

B)

THE DIFFERENCE BETWEEN RATIONAL NUMBERS AND IRRATIONAL
NUMBERS.
Rational Numbers
Irrational Numbers
1
= 0.333…
3
2
= 1.41421356…
 = 3.14159265…
4
= 0.363636…
11
e  2.71828128...
1
= 0.25
4
2
Mathematics DM015
Topic 1 : Number System
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C)
RELATIONSHIP OF NUMBERS SETS [N, W , Z , Q , Q and R]
i) Represented grammatically
Q
Q
Z
W
N
From the diagram, we can see that : 1. N  W  Z  Q 
2. Q  Q =
Example 1
For the set of {-5, -3, -1, 0, 3, 8}, identify the set of
(a) natural numbers
(b) whole numbers
(d) even numbers
(e) negative integers
(c) prime numbers
(f) odd numbers
Example 2
1 2
,  , 0, 4, 5.1212…}, identify the set of
3
(a) natural numbers
(b) whole numbers
(c) integers
(d) rational numbers
(e) irrational numbers
(f) real numbers
Given S = {-9,
7,
Example 3
a
b
(b) 5.45959…
Express each of this number as a quotient
(a) 1.555…
3
Mathematics DM015
Topic 1 : Number System
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LECTURE 2 OF 7
TOPIC
: 1.0 NUMBER SYSTEM
SUBTOPIC
: 1.1 Real Numbers
LEARNING
OUTCOMES
: At the end of the lesson, students should be able to
(d)
represent open, closed and half-open intervals and their
representations on the number line.
(e)
simplify union, , and intersection, , of two or more intervals
with the aid of number line.
The Number Line
The set of numbers that corresponds to all point on number lines is called the set of real
number. The real numbers on the number line are ordered in increasing magnitude from the left
to the right
For example for –3.5,
4
2
and  can be shown on a real number line as
3
3
2
3.5
Symbol
a=b
a<b
a>b
1
0
1
2
3
Description
a equal to b
a less than b
a greater than b
Note: The symbols ‘<’ or ‘>’ are called inequality sign
4
2
3
4

Example
3=3
4 < 4
5>0
Mathematics DM015
Topic 1 : Number System
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A) REPRESENTED ON NUMBER LINE.
All sets of real numbers between a and b, where a  b can be written in the form of intervals as
shown in the following table.
Type of Interval
Notation
Inequalities
Closed interval
[a, b]
axb
Open interval
(a, b)
a<x<b
Half-open interval
(a, b]
Half-open interval
Representation on the number line
a
b
a
b
a<x  b
a
b
[a, b)
a  x<b
a
b
Open interval
(, b)
 < x < b
b
Half-open interval
(, b]
 < x  b
b
Open interval
(a, )
a<x<
a
Half- open interval
[a, )
a  x<
a
Note :
The symbol  is not a numerical. When we write [a, ), we are simply referring to the interval
starting at a and continuing indefinitely to the right.
Example 4
List the number described and graph the numbers on a number line.
(a) The whole number less than 4
(b) The integer between 3 and 9
(c) The integers greater than -3
Example 5
Represent the following interval on the real number line and state the type of the interval.
(a) [-1, 4]
(b) {x : 2  x  5}
(c) [2, )
(d) {x : x  0, x  R}
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Mathematics DM015
Topic 1 : Number System
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B) INTERSECTION AND UNION
Intersection and union operation can be performed on intervals.
For example;
A = [1 , 6) and B = (2, 4),
The Intersection of set A and set B is a half-open interval [1,4).
The union of set A and set B is given by A  B = (2, 6) is an opened interval.
All these can be shown on a number line given below:
AB
AB
B
2
A
1
4
6
Example 6
Solve the following using the number line
(a) [0, 5)  (4, 7)
(c) (, 0]  [0, )
(b) (, 5)  (1, 9)
(d) (4, 2)  (0, 4]  [2, 2)
Example 7
Given A = {x : -2  x  5} and B = {x : 0  x  7}. Show that A  B = (0, 5].
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Mathematics DM015
Topic 1 : Number System
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EXERCISES 1:
1. Given {-7, - 3 , 0,
(a) N
1 
, ,
5 2
4 , 0.16, 0.8181…}. List the numbers for
(b) Z
(c) Q
(d) R
2. Express each of there numbers as a quotient
(a) 0.444…
(b) 0.5353…
(e) Q
a
b
(c) 1.777…
3. Represent these intervals on a number line.
(a) [2, 5]
(b) (4, 5]
(c) (8, 8)
(d) [2, )
4. Write down the following solution set in an interval notation.
(a) {x : x  6}
(b) {x : 3  x < 6}
(c) {x : 5 < x  5}
(d) {x : 0  x  4}
(e) {x : x  8}  {x : x  13} (f) {x : x  4}
5. Given A = [2, 5], B = (3, 5], C = (7, 7) and D = [3, ). Find
(a) A  B
(b) A  B
(c) A  C  D
(d) C  D  A
Answers:
1.
(a) none
(b) {-7, 0,
(c) {-7, 0,
4}
1
,
5
4 , 0.16, 0.8181…}
(d) {-7, - 3 , 4 , 0,
(e) {- 3 ,
2.
(a)
4
9

}
2
(b)
53
99
1 
, , 0.16, 0.8181…}
5 2
(c)
17
9
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Mathematics DM015
Topic 1 : Number System
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3. (a)
2
5
(b)
-4
5
(c)
-8
8
(d)
-2
4.
(a) [6, )
(b) [-3, 6)
(c) (-5, 5]
(d) [0, 4]
(e) (, 8][12, ); half- open interval
(f) (-, 4] ; half-open interval
5.
(a) [2, 5]
(b) (-3, 5]
(c) [2, 5]
(d) [2, 5]
8
Mathematics DM015
Topic 1 : Number System
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LECTURE 3 OF 7
TOPIC
: 1.0 Number System
SUBTOPIC
: 1.2 Complex Numbers
LEARNING
OUTCOMES
: At the end of the lesson, students should be able to
(a) represent a complex number in Cartesian form.
(b) define the equality of two complex numbers.
(c) determine the conjugate of a complex numbers.
(d) To perform algebraic operations on complex numbers.
SET INDUCTION:
Look at this equation:
x2 + 1 = 0
This equation does not have real roots as we cannot find a value for x whenever x =   1 .This
problem was encountered by Heron Alexandra. One hundred years later, Mahavira from India
stated that a negative value does not have square root because there is no number that is squared
to produce it. In 1637, Descrates of France, introduced ‘real number’ and ‘imaginary number’.
This idea was used by Euler from Switzerland who defined it as  1 in 1948. However
‘complex number’ was introduced after one hundred years later by Gauss from Germany (1832).
CONTENT
We have already met equations such as x2 = -1 whose roots are clearly not real since when
squared they give –1 as the result. The imaginary number I, defined to be solution to the equation
x2 = -1, is a basis of this new set.
The
imaginary
unit i
Defined as
i = 1 ,
where i2 = -1
Using the imaginary unit i, we can express the square root of any negative number as a real
multiple of i. For example,
 16 = 1 16 = i 16 = 4i,
3= i 3
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Mathematics DM015
Topic 1 : Number System
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Power of i can be simplified
therefore
 1  = -1
=  1   1  = -i
=  1   1  =  1 1 =1
2
i2 =
i3
i4
2
1
2
2
Complex Numbers,
Definition:
The set of all numbers in the Cartesian form, z = a + bi with real numbers a and b, and i the
imaginary unit, is called the set of complex numbers. The real number a is called the real part,
and the real number b is called the imaginary part, of the complex numbers. If a = 0 and b ≠ 0,
then the complex number bi is called a pure imaginary number.
Cartesion form: z = a + bi
b= Imaginary part of complex
numbers
a = real part
a & b = real numbers
= Re (z)
Example 8
(a)
If z = 2 + 3i ,
(b)
If z = 4 – 2i ,
+
i = imaginary unit
= Im (z)
=
Set of complex numbers
Re(z) = 2 , Im (z) = 3
Re (z) = 4 and Im(z) = -2
If b = 0  z = a + i(0) = a so it is a real number. Real numbers, then, are a subset of complex
numbers,   C .
a) Equality of Complex Numbers.
For two complex numbers z 1 = a + bi and z 2 = c + di, z 1 = z 2 if and only a = c and
b = d.
Example 9
Find the value of m and n if z 1 = z 2 for z 1 = 3 + 2i and z 2 = m + ni.
Example 10
Find a and b for the following equations
(a) a + b + (a – b)i = 6 + 4i
(b) a + 2b + (a – b)i = 9 + 0i
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Mathematics DM015
Topic 1 : Number System
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b) Conjugate of Complex Numbers
The complex conjugate of a complex number, a + bi is a – bi, and the conjugate of a – bi is a +
bi. The multiplication of complex conjugates gives a real number.
(a + bi)(a - bi) = a2 + b2
(a – bi)(a + bi) = a2 + b2
Complex Numbers,
Conjugate of
a + bi
a - bi
a - bi
a + bi
**This fact is used in simplifying expressions where the denominator of a quotient is
complex.
Example 11
Simplify the expressions:
a) 1i
3
1+ i
4 + 7i
c)
2 + 5i
2i
d)
3i 2
b)
c) Adding and subtracting complex numbers
Complex numbers can add together by adding the real parts and then adding the imaginary parts.
You can subtract one complex number from another by subtracting the real parts and then
subtracting the imaginary parts.
So :
(a + bi) + (c + di) = (a + c) + (b + d)i
( a + bi) – (c + di) = (a – c) + (b – d)i
Example 12
Given that z = 2 + 3i and w = 7- 6i , find
(a)
z+w
(b)
w–z
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Mathematics DM015
Topic 1 : Number System
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d) Multiplying one complex number by another
To multiply two complex numbers together, apply the rules of algebra.
So :
(a + bi) (c + di) = ac + adi +bci +bdi2
= ac + (ad + bc)i – bd ( i2 = -1 )
= (ac – bd) + (ad + bc)i
Example 13
Given that
(a) z = 4 + 3i and w = 7 +5i , find zw.
(b) w = 4 – 2i and z = -7 + 5i, find zw.
e) Solving equations
Just as you can have equations with real numbers, you can have equations with complex
numbers, as illustrated in the example below.
Example 14
Solve each of the following equations for the complex number z.
4+5i = z - (1-i)
Example 15
Solve (x + yi)(3 – i) = 1 + 2i where x and y are real.
Example 16
1 i
If z =
, find z in the Cartesian form a + bi.
2i
Example 17
If z =1 – 2i, express z +
1
in the form a + bi
z
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Mathematics DM015
Topic 1 : Number System
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EXERCISE 2:
1) Write the following complex numbers in the form x + yi
a) (3 +2i) + ( 2 + 4i)
b) (4 + 3i) - ( 2 + 5i)
c) (3 + 2i)( 4 - 3i )
Ans:
a)5 + 6i
b) 2 – 2i
c) 18 - i
2. Solve for z when
a)
z( 2 + i) = 3 – 2i
b)
(z + i)(1 – i) = 2 + 3i
1
1
3
c)
+
=
z 2  i 1 i
Ans :
a)
4 7
 i
5 5
b)
1 3
 i
2 2
c)
11 17
 i
10 10
3) Find the values of the real numbers x and y in each of the following:
x
y
1
a)
+
1  i 1  2i
x
yi
2


b)
2  i i  3 1 i
Ans :
4
5
,y
3
3
4
18
b) x  , y 
5
5
a) x 
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Mathematics DM015
Topic 1 : Number System
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LECTURE 4 OF 7
TOPIC
: 1.0 NUMBER SYSTEM
SUBTOPIC
: 1.3 Indices, surds and Logarithms
LEARNING
: At the end of the lesson, student should be able to
OUTCOMES
(a)
state the rules of indices
SET INDUCTION:
The words ‘square’ and ‘cube’ come from geometry. If a square has side x units, the square of
the number x gives the area of the square. If a cube has side x units, the cube of the number x
gives the volume of the cube.
These are written as x2 and x3 .This notation can be extended to higher powers such as x4 and x5,
although there is no obvious physical interpretation of these expressions. The index notation is
used to save us having to write several multiplications.
56 means 5  5  5  5  5  5
As defined, the notation is restricted to positive whole numbers, but it can be further extended to
1
negative and fractional powers, so that we can speak of 2-3, or of 3 2 .
CONTENT:
We can write 2  2  2  2  2 as 25 .This is known as the power (sometimes called an exponent
or index); a to the power x is written as ax .So ax is an expression in which a is the base and x is
the power.
INDICES
Rules of indices
1.
am x an = a(m+n)
2.
(am ) n = a m x n = a mn
am
m
n
3.
a  a = n = a m n
a
4.
a 0 = 1 provided a  0
1
5.
a- m = m
a
power
x
a
base
1
6.
7.
8.
9.
am =
m
n
m
n
a
m
1
n
a = a = (a ) m
(ab ) n = an bn
n
 an 
a
    n 
b
b 
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Mathematics DM015
Topic 1 : Number System
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There are some special cases to consider.
1. Division when powers are equals (Zero index)
63  63 = 63 - 3
1= 60
This result leads to another rule.
a 0 = 1 provided a  0
2. Division when the power of the denominator is greater than that of the numerator
(Negative index)
73  75 = 73 – 5
777
= 7-2
77777
1
= 7-2
72
a- m =
1
am
A negative power indicates a reciprocal.
3. Rational index
Rule 1 can be used to introduce rational power.
1
Suppose that p = q = in rule 1
2
1
1 1

2
1
a2 x a2=a2
1
2
(a ) 2 = a 1 = a

1
2
= a
1
The meaning of power can be established in a similar way.
3
a
1
3
1
3
a x a x a
1
3
(a ) 3
1
3
= a
1 1 1
 
3 3 3
= a1 = a
1
a3 =
3
a
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Mathematics DM015
Topic 1 : Number System
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1
1 1
This means ‘power ’ implies a cube root. Investigate powers , , and so on in the same way.
3
4 5
Notice how the fraction is related to the root.
a
1
m
=
m
a
Example 18
Simplify:
35  3 6
(a)
34
(b)
18 x 2 y 5
3x 4 y
3 p q r 
2
(c)
4
3
Example 19
Write the following expressions in index form:
a)
2
x3
c) x 4 2
1
1
2x 4
b)
d)
3
54 x 4
2x
Example 20
Without using calculator, evaluate:
(a)
9

3
2
(b)
 11 
1 
 25 
Example 21
Simplify the following expression

3
 a 2 b  3   x  2 b 1 
 1 2   3 1 
(a)
 x y   a2 y3 



1
2
0.04 2
3
(c)
2 2 n 4  24.2 2n1
(b)
 
10 2 n
2
EXERCISE 3:
1.
Find the value of each of the following
a) 64
2
3
1
 27  3
b)  
 8 
c)
(0.04)
3
2
d)
2 n  8 3n  4 5 n
Answer
a) 16
b)
2
3
c) 125
d) 1
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Mathematics DM015
Topic 1 : Number System
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LECTURE 5 OF 7
TOPIC
: 1.0 NUMBER SYSTEM
SUBTOPIC
: 1.3 Indices, surds and Logarithms
LEARNING : At the end of the lesson, student should be able to
OUTCOMES
(a)
Explain the meaning of a surd and its conjugate, and to carry out
algebraic operations on surds.
SET INDUCTION
A calculator will give 2 as 1.414213562, to 10 significant figures. This is not the exact value,
because 2 is irrational, and it can never be represented exactly as a fraction or as terminating
decimal.
A number expressed in terms of root sign is a surd or radical. There are many advantages in
expressing the number in this way, instead of writing it as a decimal.
CONTENT:
Expression of
Root sign
Radical
exp :
Surd
4 2
exp :
27  3
3
#Answer equal to whole number
Some examples of surds are 2 ,
approximation is precise.
7
5 and
71
#Answer equal to irrational number
7 . They cannot be evaluated exactly, i.e. no
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Mathematics DM015
Topic 1 : Number System
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A) RULES OF SURDS
a ×
ab =
a
b
b
a
=
b
,
a, b  0
PROPERTY 1
,
a,  0 , b  0
PROPERTY 2
a b + c b = (a + c) b
PROPERTY 3
a b - c b = (a - c) b
PROPERTY 4
B) ADDITIONAL RULES
1)
a a a
2)
a +
3)
a 
4)
( a +
b ) 2 = a + b+ 2 ab
5)
( a +
b )( a -
a = 2 a
b =
a
b
b ) = a2 – b2
Remark:
a b

( a +
b )
b)
24
Example 22
Simplify:
45
a)
c)
6 7 2 7
C) Multiplying Radicals
There are 2 types.
1) a  b = ab
2) a b  c d = ac bd
18
d)
5 3  27
Mathematics DM015
Topic 1 : Number System
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Example 23
Multiply:
1.
3 6 5 7
3.
7 8  10 6
3 (4 7 - 3 )
4.
8 2 (5 6 +
5.
(2 3 +4 2 )(6 3 +2 2 )
2.
2)
Example 24
Expand and simplify ( 8  3 ) ( 8  3 ) .
The conjugate of
number.
a +
For example, ( 8 +
b is
3 )( 8 -
a –
b )( a –
b where ( a +
b ) = a – b a rational
3 ) = ( 8 )2 - ( 3 )2 = 8 – 3 = 5
D) Rationalising the denominator
When the denominator of a fraction contains a square root, we generally simplify the expression
by rationalising the denominator.
To rationalize a denominator, multiply the numerator and the denominator of the fraction that
will result in the denominator to become a rational number.
If Denominator
Contains the Factor
3
To Obtain Denominator
Free from surds
Multiply by




3
3 1
3 1
2 3
2 3
 3
3 1  3 1  2
2   3  2  9  7
5   3  5  3  2
3
2
2
2
2
2
2
2
5 3
5 3
In rationalizing the denominator of a quotient, be sure to multiply both the numerator and the
denominator by the same expression.
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Mathematics DM015
Topic 1 : Number System
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Example 25
Rationalise:
2 3
3
5
1
(a)
(b)
(c)
(d)
5 3
2 3
3
7 2
Example 26
Rationalise the denominator and simplify.
1 2 1 3
17  5

(a)
(b)
1 2 1 2
17  5
(c)
E) Solving Surd Equation
Example 27
Solve each of the following equation :
2x  1  5  0
(a)
(c)
3x  1 + 1 = x
x + x2 =2
(d)
x  13  7  x  2
(e)
3x  1  2 x  1  x  2
(b)
EXERCISE 4:
1.
Simplify
63
3
75  2 48  5 12
(a)
(b)
Answer :
2.
(a)
7
(b)
3 3
Expand and simplify:
(a)
(2 - 3 3 )(3 + 2 3 )
(b)
(5 - 2 7 )(5 + 2 7 )
Answer :
(a)
-12 - 5 3
(b) -3
20
3 2
3 2
 2
Mathematics DM015
Topic 1 : Number System
______________________________________________________________________________
LECTURE 6 OF 7
TOPIC
: 1.0 NUMBER SYSTEM
SUBTOPIC
: 1.3 Indices, surds and Logarithms
LEARNING : At the end of the lesson, student should be able to
OUTCOMES
(a)
state and prove the laws of logarithm such as loga MN = loga M +
M
loga N, loga
= loga M - loga N and loga MN = N loga M.
N
SET INDUCTION:
The Richter scale measures the magnitude of earthquakes using logarithms. The Richter value is
only part of the number describing on earthquake is 106. Many people believe an earthquake
with a Richter value of 8 is twice as strong as one with value of 4. How do the two actually
compare in intensity?
CONTENT
A) Common Logarithms
Logarithm is a number y (y > 0) for any given base a (a > 0) and is written as log a y  x where
y= a x and x  R.
B) Natural Logarithms
The natural logarithmic function and the natural exponential function are inverse functions of
each other. The symbol ln x is an abbreviation for loge x, and we refer to it as the natural
logarithms of x.
Definition of Common Logarithms
log x = log10x
,
for every x > 0
Definition of Natural Logarithms
ln x = loge x , for every x > 0
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Mathematics DM015
Topic 1 : Number System
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Example 28
1. 2 3 = 8
thus 3 is the power to which the base 2 must be raised to obtain 8 or 3 is the logarithm
which, with a base 2, gives 8.
This is written simply as 3 = log 2 8
2.
3.
32 = 9
thus 2 is logarithm which, with a base 3, gives 9 or 2 = log 3 9
1
( ) 2 = 25
5
1
, gives 25 or -2 = log 1 25
5
5
So, we see that the base of a logarithm may positive be any number. Common logarithms have
base 10. From a calculator it is found that the logarithm of 5 is 0.6990
thus -2 is logarithm which, with a base
therefore
10 0.6990 = 5
or log 10 5 = 0.6990
It is usual to omit the base 10 and to write simply
lg 5 = 0.6990 or log 5 = 0.6990
But any base other than 10 must be stated.
Therefore
log 100 = 2
(i.e
100 = 10 2 )
And
log 0.1 100 = -2
(i.e
100 = 0.1 2 )
In general,
log a b = c  b = a c
Example 29
For each of the following, write down an expression for a logarithm in a suitable base:
1
1
(a)
100 = 10 2
(b)
= 5
32
2
Example 30
Without using calculator, find the value of:
1
log 1 (4)
(a)
log2  
(b)
8
2
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Mathematics DM015
Topic 1 : Number System
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(C) The Laws of Logarithms
Let
Therefore
log a b = x
ax = b
and
log a c = y
and
ay = c
a x . a y = bc
or
a x y
therefore
x + y = log a bc
i.e
Similarly
= bc
log a b + log a c = log a bc
ax ay
b
c
=
b
c
Or
a x y
therefore
x - y = log a
b
c
log a b - log a c = log a
b
c
i.e
Let
log a b n
RULE 1
=
RULE 2
= z
Therefore
az
So
an
= bn
z
Therefore
log a b
=b
=
z
n
Or
n log a b
= z
i.e
n log a b
= log a b n
RULE 3
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Mathematics DM015
Topic 1 : Number System
______________________________________________________________________________
The rule 1,2 and 3 are known as the three basic laws of logarithms, i.e.
1. Product rule : log a MN = log a M + log a N
2. The Quotient rule: log a
M
= log a M - log a N
N
3. The power rule: log a M N = N log a M
The following table lists the general properties for natural logarithmic form.
Logarithms with base a
Common logarithms
Natural logarithms
1) loga(1) = 0
log 1 = 0
ln 1 = 0
2) logaa = 1
log 10 = 1
ln e = 1
3) logaax = x
log 10x = x
ln ex = x
4) a loga ( x )  x
10 log(x )  x
e ln(x )  x
Example 31
Expand:
(a)
ln 4x
e3
ln
(c)
7
(b)
log 10x
(d)
ln x
Example 32
Given log 2 = 0.301 and log 6 = 0.778 find log 12.
Example 33
Given log 6 = 0.778, find log 36.
Example 34
Write the following as single logarithms:
a)
log 8 – log 6 + log 9
2 ln (x + 7) – ln x
b)
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Mathematics DM015
Topic 1 : Number System
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(D) Properties of Logarithms
The properties of logarithms are very similar to the properties of exponents because logarithms
are exponents. The properties will be used in solving logarithmic equations.
Properties of logarithms
Example
1) log a a  1
log5(5) = 1
2) loga(1) = 0
log2(1) = 0
3) loga(am) = m
log (107) = 7
4) a loga ( m )  m
5 log5 ( 3)  3
1
5) log a     log a ( N )
N
1
log    log 1  log 3
3
  log 3
6) loga(m) = loga(n)  m=n
log x = log 4  x = 4
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Mathematics DM015
Topic 1 : Number System
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LECTURE 7 OF 7
TOPIC
: 1.0 NUMBER SYSTEM
SUBTOPIC
: 1.3 Indices, surds and Logarithms
LEARNING : At the end of the lesson, student should be able to
OUTCOMES
(a)
change the base of logarithm using, loga M =
log b M
.
log b a
(E) Change of Base
Tables are not readily available which list the values of expressions such as log 7 2.
However if
log 7 2 = x
7x = 2
x log 7 = log 2
log 2
Or
x =
log 7
Therefore we have changed the base from 7 to 10 and can now use our ordinary tables or
calculator to evaluate log 7 2.
Then
So
Therefore
log 7 2 =
0.3010
log 2
=
= 0.3562
0.8451
log 7
A general formula for changing from base a to base b can be derived in the same way.
If
log a c = x
Then
So
c = ax
log b c = x log b a

x=
log b c
log b a
or
log a c 
log b c
log b a
in the special case when c = b , this identity becomes
log a b 
log b b
log b a
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Mathematics DM015
Topic 1 : Number System
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or
1
log a b 
log b a
Example 35
Find the following expression to four decimal places.
(a)
log35
(b)
log510
Example 36
Find x if ln x = 4.7
Example 37
Change to logarithmic form for the expression e7 = p
EXERCISE 5:
1.
Without using calculator, find the values of:
(a)
log 1000
(b)
log 0.01
2.
Find the values of the following, to three significant figures:
1
(a)
log 50
(b)
log( )
3
3.
Without using calculator, find the exact values of x in:
(a)
4.
x = log 2 64
(b)
x = log 3 (
1
)
81
Find the values of the a
(a)
3 = log a 125
(b)
-2 = log a (
27
1
)(c)
25
1
= log a 25
2