Drawing fractal by hand is tedious at best and close to impossible at worst. With the help of a CAS program, all you need is a few lines of code with a for-loop operation.
The fractal structure in Fig. 1 is generated from Gumnowski-Mira chaos model
by Omega CAS Explorer.
Replace the term by with b = 0.9998, the result is shown in Fig. 2.
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.
However, a more explicit expression is better suited for this purpose.
From Fig.2, geometrical Interpretation of as the shaded area reveals that
Inserting into (1) the well known result
We conclude that
As a consequence,
(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),
Subtracting (3) from (2) and using the fact that , we have
Since this solution can be expressed as
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)