In this unit we are going to be looking at logarithms. However, before we can deal with logarithms
we need to revise indices. This is because logarithms and indices are closely related, and in order
to understand logarithms a good knowledge of indices is required.
We know that
16 = 24
Here, the number 4 is the power. Sometimes we call it an exponent. Sometimes we call it an
index. In the expression 24 , the number 2 is called the base.
We know that 64 = 82 .
In this example 2 is the power, or exponent, or index. The number 8 is the base.
2. Why do we study logarithms ?
In order to motivate our study of logarithms, consider the following:
we know that 16 = 24 . We also know that 8 = 23
Suppose that we wanted to multiply 16 by 8.
One way is to carry out the multiplication directly using long-multiplication and obtain 128.
But this could be long and tedious if the numbers were larger than 8 and 16. Can we do this
calculation another way using the powers ? Note that
24 × 23
can be written
16 × 8
using the rules of indices which tell us to add the powers 4 and 3 to give the new power, 7. What
was a multiplication sum has been reduced to an addition sum.
Similarly if we wanted to divide 16 by 8:
16 ÷ 8
24 ÷ 23
can be written
using the rules of indices which tell us to subtract the powers 4 and 3 to give the new power, 1.
If we had a look-up table containing powers of 2, it would be straightforward to look up 27 and
obtain 27 = 128 as the result of finding 16 × 8.
Notice that by using the powers, we have changed a multiplication problem into one involving
addition (the addition of the powers, 4 and 3). Historically, this observation led John Napier
(1550-1617) and Henry Briggs (1561-1630) to develop logarithms as a way of replacing multiplication with addition, and also division with subtraction.