Let us turn our attention to the numerical calculation of logarithm, introduced in my previous post “Introducing Lady L“.

An example of naively compute , based solely on its definition is shown in Fig. 1.

Fig. 1

However, a more explicit expression is better suited for this purpose.

Fig. 2

From Fig.2, geometrical Interpretation of as the shaded area reveals that

,

i.e.,

Inserting into (1) the well known result

,

we obtain

.

Let

,

we have

.

If ,

otherwise (

.

Therefore, either

or

.

Since ,

and

We conclude that

.

As a consequence,

,

i.e.,

(2) offers a means for finding the numerical values of logarithm. However, its range is limited to the value of between 0 and 2, since .

To overcome this limitation, we proceed as follows:

. By (2),

i.e.,

Subtracting (3) from (2) and using the fact that , we have

.

i.e.,

Solving equation

where ,

we find

.

Since this solution can be expressed as

or

.

It shows that for any , . Therefore, (4) can be used to obtain the logarithm of *any* positive number. For example, to obtain , we solve first and then compute a partial sum of (4) with sufficient large number of terms (see Fig. 3)

Fig. 3