Chapter 3
Logarithms
3.1 Introduction to logarithms
With the use of calculators firmly established, logarithmic tables are now rarely used for calculation. However,
the theory of logarithms is important, for there are several scientific and engineering laws that involve the rules
of logarithms.
From the laws of indices:
16 = 2
4
The number 4 is called the power or the exponent or
the index. In the expression 2 4 , the number 2 is called
the base.
In another example:
64 = 8
2
In this example, 2 is the power, or exponent, or index.
The number 8 is the base.
What is a logarithm?
Consider the expression 16 = 2 4 .
An alternative, yet equivalent, way of writing this
expression is: log 2 16 = 4.
This is stated as ‘log to the 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 = 2 4 and log 2 16 = 4 are
equivalent.
If we write either of them, we are automatically implying the other.
In general, if a number y can be written in the form
a x , then the index ‘x’ is called the ‘logarithm of y to the
base of a’,
i.e.
if y = a
x then x = log a y
In another example, if we write down that 64 = 8 2
then the equivalent statement using logarithms is:
log 8 64 = 2
In another example, if we write down that: log 3 81 =4
then the equivalent statement using powers is:
3
4
= 81
So the two sets of statements, one involving powers
and one involving logarithms, are equivalent.
Common logarithms
From above, if we write down that: 1000 = 10 3 , then
3 = log 10 1000
This may be checked using the ‘log’ button on your
calculator.
Logarithms having a base of 10 are called common
logarithms and log 10 is often abbreviated to lg.
The following values may be checked by using a
calculator:
lg 27.5 = 1.4393 ..., lg 378.1 = 2.5776 ...
and lg 0.0204 = −1.6903 ...
Napierian logarithms
Logarithms having a base of e (where ‘e’ is a mathematical constant approximately equal to 2.7183) are
called hyperbolic, Napierian or natural logarithms,
and log e is usually abbreviated to ln.
The following values may be checked by using a
calculator:
ln 3.65 = 1.2947 ..., ln 417.3 = 6.0338 ...
and ln 0.182 = −1.7037 ...
More on Napierian logarithms is explained in Chapter 4
following.
Here are some worked problems to help understanding of logarithms.
Logarithms
3.1 Introduction to logarithms
With the use of calculators firmly established, logarithmic tables are now rarely used for calculation. However,
the theory of logarithms is important, for there are several scientific and engineering laws that involve the rules
of logarithms.
From the laws of indices:
16 = 2
4
The number 4 is called the power or the exponent or
the index. In the expression 2 4 , the number 2 is called
the base.
In another example:
64 = 8
2
In this example, 2 is the power, or exponent, or index.
The number 8 is the base.
What is a logarithm?
Consider the expression 16 = 2 4 .
An alternative, yet equivalent, way of writing this
expression is: log 2 16 = 4.
This is stated as ‘log to the 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 = 2 4 and log 2 16 = 4 are
equivalent.
If we write either of them, we are automatically implying the other.
In general, if a number y can be written in the form
a x , then the index ‘x’ is called the ‘logarithm of y to the
base of a’,
i.e.
if y = a
x then x = log a y
In another example, if we write down that 64 = 8 2
then the equivalent statement using logarithms is:
log 8 64 = 2
In another example, if we write down that: log 3 81 =4
then the equivalent statement using powers is:
3
4
= 81
So the two sets of statements, one involving powers
and one involving logarithms, are equivalent.
Common logarithms
From above, if we write down that: 1000 = 10 3 , then
3 = log 10 1000
This may be checked using the ‘log’ button on your
calculator.
Logarithms having a base of 10 are called common
logarithms and log 10 is often abbreviated to lg.
The following values may be checked by using a
calculator:
lg 27.5 = 1.4393 ..., lg 378.1 = 2.5776 ...
and lg 0.0204 = −1.6903 ...
Napierian logarithms
Logarithms having a base of e (where ‘e’ is a mathematical constant approximately equal to 2.7183) are
called hyperbolic, Napierian or natural logarithms,
and log e is usually abbreviated to ln.
The following values may be checked by using a
calculator:
ln 3.65 = 1.2947 ..., ln 417.3 = 6.0338 ...
and ln 0.182 = −1.7037 ...
More on Napierian logarithms is explained in Chapter 4
following.
Here are some worked problems to help understanding of logarithms.