Chapter 15

Logarithms

15.1 Introduction to logarithms

With the use of calculators firmly established, logarith-
mic tables are now rarely used for calculation. However,
the theory of logarithms is important, for there are sev-
eral scientific and engineering laws thatinvolve the rules
of logarithms.

From Chapter 7, we know that 16 = 24.

The number 4 is called thepoweror theexponentor
theindex. In the expression 2^4 , the number 2 is called
thebase.

In another example, we know that 64 = 82.

In this example, 2 is the power, or exponent, or index.
The number 8 is the base.

15.1.1 Whatis a logarithm?

Consider the expression 16= 24.
An alternative, yet equivalent, way of writing this
expression is log 216 =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 that becomes the base of the
logarithm.

The two statements 16= 24 and

log 216 =4 are equivalent

If we write either of them, we are automatically imply-
ing the other.
In general, if a numberycan be written in the formax,
then the indexxis called the ‘logarithm ofyto the base

ofa’, i.e.

ify=axthenx=logay

In another example, if we write down that 64= 82 then

theequivalent statement usinglogarithmsislog 864 =2.

In another example, if we write down that log 327 = 3

then the equivalent statement using powers is 3^3 =27.

So the two sets of statements, one involvingpowers and

one involving logarithms, are equivalent.

15.1.2 Common logarithms

From the above, if we write down that 1000= 103 ,then

3 =log 10 1000. This may be checked using the ‘log’

button on your calculator.

Logarithms having a base of 10 are calledcommon

logarithmsand log 10 is usually abbreviated to lg. The

following values may be checked using a calculator.

lg27. 5 = 1. 4393 ...

lg378. 1 = 2. 5776 ...

lg0. 0204 =− 1. 6903 ...

15.1.3 Napierian logarithms

Logarithms having a base ofe(whereeis a mathemat-

ical constant approximately equal to 2.7183) are called

hyperbolic, Napierian ornatural logarithms,and

logeis usually abbreviated to ln. The following values

may be checked using a calculator.

ln3. 65 = 1. 2947 ...

ln417. 3 = 6. 0338 ...

ln0. 182 =− 1. 7037 ...

Napierian logarithms are explained further in Chapter

16, following.

DOI: 10.1016/B978-1-85617-697-2.00015-6