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