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Logarithms
Logarithms appear in all sorts of calculations in engineering and science, business and economics.
Before the days of calculators they were used to assist in the process of multiplication by replacing
the operation of multiplication by addition. Similarly, they enabled the operation of division to
be replaced by subtraction. They remain important in other ways, one of which is that they
provide the underlying theory of the logarithm function. This has applications in many fields, for
example, the decibel scale in acoustics.
In order to master the techniques explained here it is vital that you do plenty of practice exercises
so that they become second nature.
After reading this text and / or viewing the video tutorial on this topic you should be able to:
• explain what is meant by a logarithm
• state and use the laws of logarithms
• solve simple equations requiring the use of logarithms.
Contents
1.
Introduction
2
2.
Why do we study logarithms ?
2
3.
What is a logarithm ?
4.
Exercises
5.
The first law of logarithms
loga xy = loga x + loga y
4
6.
The second law of logarithms
loga xm = m loga x
5
7.
The third law of logarithms
loga
8.
The logarithm of 1
loga 1 = 0
9.
Examples
6
10. Exercises
8
11. Standard bases 10 and e
if x = an then loga x = n
3
4
x
y
= loga x − loga y
log and ln
12. Using logarithms to solve equations
5
6
8
9
13. Inverse operations
10
14. Exercises
11
1

1. Introduction
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.
Example
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
This equals
27
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
This equals
21
or simply
2
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.
2

3. What is a logarithm ?
Consider the expression 16 = 24 . Remember that 2 is the base, and 4 is the power. An alternative,
yet equivalent, way of writing this expression is log2 16 = 4. This is stated as ‘log to base 2 of 16
equals 4’. We see that the logarithm is the same as the power or index in the original expression.
It is the base in the original expression which becomes the base of the logarithm.
The two statements
16 = 24
log2 16 = 4
are equivalent statements. If we write either of them, we are automatically implying the other.
Example
If we write down that 64 = 82 then the equivalent statement using logarithms is log8 64 = 2.
Example
If we write down that log3 27 = 3 then the equivalent statement using powers is 33 = 27.
So the two sets of statements, one involving powers and one involving logarithms are equivalent.
In the general case we have:
Key Point
if x = an
then equivalently
loga x = n
Let us develop this a little more.
Because 10 = 101 we can write the equivalent logarithmic form log10 10 = 1.
Similarly, the logarithmic form of the statement 21 = 2 is log2 2 = 1.
In general, for any base a, a = a1 and so loga a = 1.
Key Point
loga a = 1
3

We can see from the Examples above that indices and logarithms are very closely related. In
the same way that we have rules or laws of indices, we have laws of logarithms. These are
developed in the following sections.
4. Exercises
1. Write the following using logarithms instead of powers
a) 82 = 64
b)
35 = 243
10−3 = 0.001 g) 3−2 = 19
√
49 = 7
k) 272/3 = 9
e)
106 = 1000000
f)
i)
5−1 =
1
5
j)
c)
210 = 1024 d)
53 = 125
h)
60 = 1
l)
32−2/5 =
1
4
2. Determine the value of the following logarithms
a)
log3 9
e)
log4 64
i)
log3
1
27
5
m) loga a
b)
log2 32
c) log5 125
d) log10 10000
f)
log25 5
g) log8 2
h) log81 3
j)
n)
log7 1
√
logc c
k) log8
1
8
o) logs s
l) log4 8
p) loge
1
e3
5. The first law of logarithms
Suppose
x = an
and
y = am
and
loga y = m
then the equivalent logarithmic forms are
loga x = n
(1)
Using the first rule of indices
xy = an × am = an+m
Now the logarithmic form of the statement xy = an+m is loga xy = n + m. But n = loga x and
m = loga y from (1) and so putting these results together we have
loga xy = loga x + loga y
So, if we want to multiply two numbers together and find the logarithm of the result, we can do
this by adding together the logarithms of the two numbers. This is the first law.
Key Point
loga xy = loga x + loga y
4

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