Download Unit 1 Indices and logarithms(student)

CMV6120
Foundation Mathematics
Unit 1
Indices and logarithms
Learning Objectives
The students should be able to:

Use the laws of indices to solve simple problems.

Use the properties of logarithms to solve simple problems.
Unit 1:Indices and Logarithms
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CMV6120
Foundation Mathematics
Indices
In dealing with expressions containing exponents along with addition or subtraction or
multiplication or division, we work with the exponents first. Lets consider the following
examples.
Example 1 (a)
3 2  2 3  _________
(b)
2 4  5 3  16  125 = ___________
(c)
3 x 10 4  3 x _______ = __________
Example 2 Write each of the following in another way by using exponents.
(a) (ab)(ab)(ab)
(b) –a.a.a.a
(c) 4.a.b.4.b.a.a.a
Solution:
Example 3 Evaluate each of the following :
(a) -34
(b) (-3)4
(c) 2(1.1)3
Solution:
Unit 1:Indices and Logarithms
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CMV6120
1.
Foundation Mathematics
Multiplication of Exponential Numbers
Consider
23.24 = (2.2.2)(2.2.2.2)
= 2.2.2.2.2.2.2
= 27
a2.a3 = (a.a)(a.a.a)
= a.a.a.a.a
= a5
am.an = am+n
In general,
2.
Division of Exponential Numbers
Consider
4 7 4.4.4.4.4.4.4

4.4.4
43
=44
a 5 a.a.a.a.a

a.a.a
a3
= a2
In general ,
am
 a mn
n
a
Example 4
Simplify the following :
319
a) 11  31911  ________
3
X9
b)
 _________
X3
a10b 8
c)
 a 103b 85  ________
3 5
a b
an
 a nn
an
= a0………………..(1)
On the other hand
an
 1............................(2)
an
Compare (1) & (2):
a0 = 1
Consider
where a is any real number and a  0
Unit 1:Indices and Logarithms
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CMV6120
Foundation Mathematics
For examples,
20 = 1,
(-3)0 = 1 and
100 = 1
am
 a m  n ………………(  )
an
23
222
1


4
2222 2
2
Recall
Consider
If we use the division rule (  )
23
 2 34  2 1
4
2
1

2-1=
2
a-n = a0-n =
In general
a-n =
Consider
34
3333
1

 2
6
333333 3
3
If we use the division rule (  )
34
 3 46  3  2
6
3
1
 3-2 = 2
3
a0
an
1
an
n
n
n
n
(a b) = a b
Example 5
(a)
an
a
   n
b
b
(am)n = amn
Simplify the following :
(a2 b)3
(b)
(
x3 2 3 2
) (y)
y
Solution :
Unit 1:Indices and Logarithms
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CMV6120
Foundation Mathematics
Example 6
Carry out each of the following operation. Write all answers with
positive exponents and simplify where possible.
(a) 7-1
(b) 2-4
(c) 5(5-3)
(d)
2 5
23
Solutions:
(a)
(c)
1
7
5(5-3) =51  5-3
7 1 
=
=
3.
1
1
or
4
16
2
(b)
2 4 =
(d)
2 5

23
=
Radicals
If x2 = y , then x is a square root of y
for example 72 = 49
∴ 7 = 2 49
similarly,
81
or
= 9
49
7=
(∵ 92=81 )
If x3 = y , then x is a cube root of y
for example, 43 = 64
(∵
3
64  4 )
In general, if xn = y , where n is a positive integer, then x is a nth root of y.
for example, 24=16
∴ 4 16  2
Unit 1:Indices and Logarithms
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CMV6120
Foundation Mathematics
Example7
(a)
(b)
(c)
Find the values of the following :
6
64
5
100000
4
81
Solution :
144=9  16
144  12
9  16  3  4
=12
∴ 144  9 16
Consider
and
9  16  3  4
In general
ab  a b
Note that
ab  a  b
e.g.
a
a

b
b
25 = 16  9
16  9  4  3  7
but
25  5
∴ 16  9  16  9
In general
Example 8
(a)
(b)
n
a na

b nb
n
ab  n a n b
Find the values of the following functions :
x
f(x) =
when x = 25
64
f(x) = x0.5
when x = 0.027
Unit 1:Indices and Logarithms
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CMV6120
Foundation Mathematics
Solution :
25
25

64
64
=__________
3
27
27
f(0.027) = 3
3
1000
1000
=_________
(a)
f(25) =
(b)
4.
Fractional Indices
n
1
n
 1n 
a   a n
 
 
=a1 = a
Consider
1
∴
ann a
n
am
Similarly ,
am/n = ( n a )m =
Example 9
Evaluate the following:
(a)
81
3
4
(b)
1000
1
2
3
(c)
 9 2
 
 25 
Solutions:
3
4
4
3
(a)
81 = ( 81 ) = __________
(c)
9
 9 2
=_______
  =
25
 25 
(b)
2
3
1000 = ( 3 1000 )2 = ________
1
Unit 1:Indices and Logarithms
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CMV6120
Example 10
Foundation Mathematics
In the compound interest formula A = P (1+r)n , if A = 54874.32 ,
P = 25000, n = 6, find the value of r .
Solution:
A = P ( 1+r)n
54874.32 = ( 25000)( 1 + r )6
54874.32
6
 1  r 
25000
54874.32
1  r  6
25000
r = ____________%
5.
Definition of Logarithms
If a number X = ay, where a is positive and a≠1, the index y is called the
logarithm of the number X to the base a. In symbol, y = log aX.
N.B. logaX is undefined only for positive values of X
For example,
23 = 8
 log28 = 3
34 = 81
 log381 = 4
When the base a is not stated in logaX, it may be assumed a = 10. This is called the
common logarithm.
For example,
log 1000 = log101000
=3
log 0.01 = -2
 1 
( 10-2 =  2  = 0.01)
 10 
Unit 1:Indices and Logarithms
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CMV6120
6.
Foundation Mathematics
Properties of Logarithms :
1.
logaa = 1
2.
loga1 = 0
3.
logaMN = logaM + logaN
4.
loga
5.
logaXn = n logaX
M
N
= logaM
- logaN
Example 11 Find the values of the following:
1
(a) log711 + Log7  
 11 
(b) log6 – log60
(c) log5125
Solution:
1
(a) log711 + log7  
 11 
1

log7 11   = log71= __________
11 

(b) log 6 – log 60
log
6
1
= log =____________
60
10
(c) log5125
= log553
= 3 log55
=______________
Unit 1:Indices and Logarithms
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CMV6120
Example 12
Foundation Mathematics
In the compound interest formula A = P ( 1 + r )n , if A = 20000,
P = 10000,
r = 12%, find n correct to 2 decimal places.
Solutions:
A = P ( 1 + r )n
20000 = 10000 ( 1 + 12% )
20000
 (1  0.12) n
10000
2 = ( 1.12 )n
Taking logarithm on both sides,
log2 = log(1.12)n
log2 = n log 1.12
n 
log 2
log 1.12
n = _________
Unit 1:Indices and Logarithms
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