Once again
we start with the general expression.

log n – log
(n – 1) = 1/n + 1/2n

^{2}+ 1/3n^{3 }+ 1/4n^{4 }+….
If we let n
= – 1, then n – 1 = – 2.

Therefore, log
n – log (n – 1) = log (– 1) – log (– 2) = log (– 1/– 2) = log (1/2)

And through
the formula expansion,

log (1/2) = – 1 + 1/2 – 1/3 + 1/4 –…,

= – (1 – 1/2
+ 1/3 – 1/4 +…) = – log 2 = – .693147…

What is
interesting here is that we can use the logs of two negative numbers, to derive
the well known log of a positive.

Now through
Euler’s Identity,

e

^{iπ}= – 1, so that log (– 1) = iπ.
And as log (–
1) – log (– 2) = log (1/2),

iπ – log (–
2) = log (1/2),

so that log
(– 2) = iπ – log (1/2),

= iπ + log 2.

More
generally, we can therefore express the log of any negative number i.e. log (– n),
through the complex expression a + it, where a = log n and t = π.

There is a
very important point that needs to be made at this point.

Just as in
conventional (Type 1) terms we can give the customary analytic interpretation of
such mathematical symbols in a quantitative manner, equally in the - as yet - unrecognised
(Type 2) terms we can give these symbols a unique holistic interpretation
in a qualitative manner.

So the fact
that the log of a negative number must be expressed in a complex mathematical fashion (with real and imaginary parts) implies in holistic terms that we properly require
here the intermingling of both quantitative interpretation (with respect to the
real part) and a corresponding qualitative interpretation (with respect to the
imaginary part).

This is a
matter over which great confusion presently exists with respect to the standard
conventional interpretation of complex logs.

So e

^{iπ}= – 1.
Therefore
squaring both sides,

e

^{2iπ }= 1
This
implies - from the conventional (Type 1) quantitative perspective - that when
we multiply the expression on the RHS by e

^{2iπ}, that the value remains unchanged as 1.
Therefore,

e

^{2iπ }= e^{4iπ }= e^{6iπ }= e^{8iπ }=….. = 1.
However
from the Type 2 perspective it looks very different!

So properly, e

^{2i}^{π}= 1^{1}, e^{4i}^{π }= 1^{2}, e^{6i}^{π }= 1^{3}, e^{8i}^{π }= 1^{4}, and so on.
Therefore though the value of these expressions does indeed
remain unchanged in a (reduced) quantitative manner according to standard Type
1 interpretation, this value continually changes with respect to the dimensional
number involved which - relatively - should be interpreted in a qualitative Type
2 manner.

A coherent interpretation therefore of the nature of the complex
behaviour of the logs of negative numbers therefore ultimately requires both analytic
(Type 1) and holistic (Type 2) interpretation, which inherently requires a
dynamic interactive manner of understanding.

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