Chapter 3

Logarithms

3.1 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 that involve the rules
of logarithms.

From the laws of indices: 16 = 24

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

In another example: 64 = 82

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= 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 which becomes the base of the
logarithm.

The two statements: 16= 24 and log 216 =4are
equivalent.
If we write either of them, we are automatically imply-
ing the other.
In general, if a numberycan be written in the form
ax, then the index ‘x’ is called the ‘logarithm ofyto the
base ofa’,

i.e. ify=ax then x=logay

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

then the equivalent statement using logarithms is:

log 864 = 2

Inanother example, ifwe writedownthat: log 381 = 4

then the equivalent statement using powers is:

34 = 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= 103 ,then

3 =log 101000

This may be checked using the ‘log’ button on your

calculator.

Logarithms having a base of 10 are calledcommon

logarithmsand log 10 is often abbreviated to lg.

The following values may be checked by using a

calculator:

lg27. 5 = 1. 4393 ..., lg378. 1 = 2. 5776 ...

and lg0. 0204 =− 1. 6903 ...

Napierian logarithms

Logarithms having a base of e (where ‘e’ is a math-

ematical constant approximately equal to 2.7183) are

calledhyperbolic,Napierianornatural logarithms,

and logeis usually abbreviated to ln.

The following values may be checked by using a

calculator:

ln3. 65 = 1. 2947 ...,ln417. 3 = 6. 0338 ...

and ln0. 182 =− 1. 7037 ...

More on Napierian logarithms is explained in Chapter 4

following.

Here are some worked problems to help understand-

ing of logarithms.