Aunc.1961
net.math
utzoo!decvax!duke!unc!bts
Sat Mar 13 20:28:32 1982
log(0) revisited
There is a problem with the argument that log 0 - log 0 = 0, unfortunately.
It is just as easy to take limits as follows:

	Lim log(a) - Lim log(b) =
	a -> 0       b -> 0			???

There are many cases where you'll get into trouble if you assume that two
variables are going to 0 at the same rate.  As far as ways to deal with log
of 0, I rather like infinitesimals.  Replace the 0's with a and b, where
a and b are "infinitely close" to 0.  Then the algebra works, but you won't
necessarily get 0 or even another infinitesimal as the answer.  Adding a
single point at infinity looks nice topologically, but you get into trouble
using that point algebraically; it won't obey the same rules as "finite"
points.  A good source for non-standard analysis is the first few chapters
of Stroyan and Luxemburg's text. (Oh yes, if you replace both occurrences
of 0 with the same infinitesimal, you do get 0.)