Indices and Logarithms

1. CHAPTER 5
INDICES AND
LOGARITHMS
What is Indices?

2. 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

3. 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

4. LAWS OF INDICES
Multiplication of indices with same base:
am  an = am + n
bm + n = bm  bn
Example:
x4 x3 = x4 + 3 = x7
y4y7 = y4+(-7) = y3 =
2x+3 = 2x  23 = 8(2x)
3y – 2 = 3y  32 =
3
y
1
 
2
1
3
3
y

5. Division of indices with same base:
am ÷ an = am  n
bm  n = bm ÷ bn
Example:
= c9  4 = c5
3x-2 =
4
9
c
c
5
2
p
12
p
4
3
p 5
2

3
p
3
1 
 3
p
3
1

2
3
3
x

6. 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

7. (ab)n = anbn
EXAMPLE:
(xy)3 = x3  y3
23  33 = 63
(ab)-2 = a-2  b-2

8. Law 5:
EXAMPLE:
 
n n
n
a a
b b

 
2 2
2
2 2
3 3

   
2 2
2
2
a a b
b a
b



 
   
2 2 2
2
5 2 2 4
2 5 25
5

  

9. Other properties of index
Zero index: a0 = 1, a  0
Negative index: a-n
Fractional index:
1
n
a

 
m m
n m n
n
a a a
 
1
n
n
a a

 
2 2
2
3
3
64 64 4
 

10. Law 5:
EXAMPLE:
 
n n
n
a a
b b

 
2 2
2
2 2
3 3

   
2 2
2
2
a a b
b a
b



 
   
2 2 2
2
5 2 2 4
2 5 25
5

  

11. Example
Solve
(a) 91 – x = 27
(b) 2p + 1  43 – p =
(c) Solve the simultaneous equation
2x.42y = 8
5x.25-y =
(d) 4x+3 – 4x+2 = 6
1
16
1
125

12. Solution
(a) x = -0.5
(b) p = 11
(c) x = -1, y = 1
(d) x = -1.5