Graphics Reference
In-Depth Information
3.4 Indices
A notation for repeated multiplication is with the use of indices. For instance,
in the above example with a quadratic equation
x
2
is used to represent
x
x
.
This notation leads to a variety of situations where laws are required to explain
how the result is to be computed.
×
3.4.1 Laws of Indices
The laws of indices can be expressed as
a
m
a
n
=
a
m
+
n
×
(3.10)
a
m
a
n
=
a
m−n
÷
(3.11)
(
a
m
)
n
=
a
mn
(3.12)
which are easily verified using some simple examples.
3.4.2 Examples
1:2
3
2
2
=8
4=32=2
5
×
×
2:2
4
4=4=2
2
3:(2
2
)
3
=64=2
6
2
2
=16
÷
÷
From the above laws, it is evident that
a
0
= 1
(3.13)
1
a
p
a
−p
=
(3.14)
√
a
p
a
q
q
=
(3.15)
3.5 Logarithms
Two people are associated with the invention of logarithms: John Napier
(1550-1617) and Joost Burgi (1552-1632). Both men were frustrated by the
time they spent multiplying numbers together, and both realized that multi-
plication could be replaced by addition using logarithms. Logarithms exploit
the addition and subtraction of indices shown in (3.10) and (3.11), and are
always associated with a base. For example, if
a
x
=
n
,thenlog
a
n
=
x
,where
a
is the base. A concrete example brings the idea to life:
if 10
2
= 100 then log
10
100 = 2
which can be interpreted as '10 has to be raised to the power (index) 2 to
equal 100'. The log operation finds the power of the base for a given number.
