1. Indices and Logarithms
What is an index ?
The idea of an index (or power) arises when we multiply a number by itself several times.
i.e. a a a a a = a5
We call the number a the base and the small number indicating the number of multiplications, the
index or power. This notation becomes a shorthand, helping us to write calculations of this sort
without multiplication signs. However, the notation soon 'takes over' and is applied to other forms
that are no longer so easy to understand. We replace the continued multiplication with a single
operation - taking an index, or raising to a power.
a5
index , power
base
By investigating what happens during the different operations of arithmetic, we can derive the laws
of indices.
Multiplication: a 2 a 3 = a a a a a
= a5
Similarly: 24 23 = 27
and t7 t11 = t18
Rule for multiplication
The law is am an = am+n when multiplying add indices
a6 a a a a a a
Division : = cancel by a twice
a2 a a
= a4

2. Rule for division
am
The law is am an or = am - n when dividing subtract indices
an
28
For example: 3
= 28 - 3 = 2 5
2
t3
4
= t 3 - 4 = t -1 but what is a negative power?
t
b 4c3
2
= b 4-2 c 3-1 = b 2 c 2
b c
Several points arise from these examples.
1. It is only meaningful to combine indices applied to the same base - hence
the powers of base b and c are subtracted separately.
2. A number or letter without a power is assigned the power of 1
i.e. c = c1 3 = 31 x = x1 etc.
3. Using the division law we can end up with both negative powers and a
power of 0, which require 'translating', so that we can understand them.
Zero Index : a 0 = 1
am
Arises in division : = a m-m = a 0
am
But dividing a number by itself gives 1 a0 = 1
Rule Any base raised to the power of zero equals 1.
1
Negative powers : a- n =
an

3. t3 t3 t t t 1
From the example above 4
= t 3 - 4 = t -1 or 4
= =
t t t t t t t
1
so t -1 = the negative power is the reciprocal
t
1
In general: a- n =
an
Rule A negative power is the reciprocal of the positive power.
1 1
For example: 2-3 = =
23 8
2 1
and =2 = 2x-3
x3 x3
It becomes a little more difficult if the negative power is in the denominator.
2x 2x
For example: = turning negative index into reciprocal
y 4 1
y4
1 y4
or 2x = 2x = 2xy4 fraction division
y4 1
In the fraction setting, a negative power becomes a positive power in the other 'part' of the fraction
- when moving from top to bottom or vice versa.
Let's look at some more examples using all the rules we have covered so far.