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CHAPTER 5
INDICES AND
LOGARITHMS
What is Indices?
Examples of numbers in index
form.
33
(3 cubed or 3 to the power of 3)
25
(2 to the power of 5)
3
and 5 are known as indices.
27=33, 3 is a base and 3 is an index
32=25, 2 is a base and 5 is an index
So , why we use indices?
Indices can make large numbers
much more manageable, as a
large number can be reduced to
just a base and an index.
Eg: 1,048,576
= 220
LAWS OF INDICES
Multiplication of indices with same base:
am  an = am + n
bm + n = bm  bn
Example:
x4 x3 = x 4 + 3 = x 7
1
4

7
4+(-7)

3
y y =y
=y = 3
y
2x+3 = 2x  23 = 8(2x)
y 1
y
–
2
y
2
3 = 3 3 = 3 2
3 
Division of indices with same base:
am ÷ an = am  n
bm  n = bm ÷ bn
Example:
9
c
9  4 = c5
=
c
4
c
x
3
x-2
3 = 2
3
25
4p 2
p
1
3
1

 p
5 
3
3
3
12 p
3p
Raising an index to a power
(am)n = amn
bmn = (bm)n
EXAMPLE:
(b4)3 = b43 = b12
(32)3 = 323 = 36
(2x)2 = 22x
(2y+1)3 = 23y + 3
32c = (3c)2
n
(ab)
=
n
n
a b
EXAMPLE:
(xy)3 = x3  y3
23  33 = 63
(ab)-2 = a-2  b-2
Law 5:

a
b
n
n
a
 n
b

a  a  b
b a
b
5  2 2  4
 2   5  5 25
EXAMPLE:
2
2
3
2
2
2
 2
3
2
2
2
2
2
2
2
Other properties of index
Zero index: a0 = 1, a  0
1
 n
a
a-n
Negative index:
Fractional index:
1
n
a  a
m
n
n
a  a 
2
3
64 
n

m
3
64
 a

n
2
4
m
2
Law 5:

a
b
n
n
a
 n
b

a  a  b
b a
b
5  2 2  4
 2   5  5 25
EXAMPLE:
2
2
3
2
2
2
 2
3
2
2
2
2
2
2
2
Example
Solve
(a) 91 – x = 27
1
p
+
1
3
–
p
(b) 2
4
= 16
(c) Solve the simultaneous equation
2x.42y = 8
1
x
-y
5 .25 =
125
(d) 4x+3 – 4x+2 = 6
Solution
(a)
(b)
(c)
(d)
x = -0.5
p = 11
x = -1, y = 1
x = -1.5