Once again we start with the general expression.
log n – log (n – 1) = 1/n + 1/2n2 + 1/3n3 + 1/4n4 +….
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,
eiπ = – 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 eiπ = – 1.
Therefore squaring both sides,
e2iπ = 1
This implies - from the conventional (Type 1) quantitative perspective - that when we multiply the expression on the RHS by e2iπ, that the value remains unchanged as 1.
e2iπ = e4iπ = e6iπ = e8iπ =….. = 1.
However from the Type 2 perspective it looks very different!
So properly, e2iπ = 11, e4iπ = 12, e6iπ = 13, e8iπ = 14, 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.