> Date: Sun, 8 Mar 2009 19:11:14 -0400 > From: Osher Doctorow Ph.D. <firstname.lastname@example.org> > To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG > Subject: Re: Primes Are Fundamentally Additive/Subtractive, > Not Multiplicative/Divisive > > To understand what is actually happening, arguably, recall that the natural logarithm is the integral from 1 to x of dt/t: > > 1) ln(x) = I[1/t)]dt, where I...dt is the integral from 1 to x with respect to t. > > Since t > 1, the argument of this integral is a number between 0 (noninclusive) and 1, quite similar and in a sense isomorphic to y/x in Conditional Probability where y is P(AB) and x is P(A) not equal to 0, or in its variant y = P(B) and x = P(A) with y < = x. So we have division fundamentally involved from an integral viewpoint in the natural logarithm and thus in all logarithms on the real numbers.
Eh? That definition for ln(x) is also valid if 0 < x <= 1 if you're restricting yourself to real variables, and x having any finite complex value except 0 is OK if not. Of course the infinitely many possibilities for the path of integration lead to the infinitely many values of the complex logarithm.
-- John Harper, School of Mathematics Statistics and Operations Research Victoria University, PO Box 600, Wellington 6140, New Zealand e-mail email@example.com phone (+64)(4)463 6780 fax (+64)(4)463 5045