Re: functional equation

From: Robert Israel (israel_at_math.ubc.ca)
Date: 10/29/04 

Date: 29 Oct 2004 19:56:11 GMT

In article <2ueq5hF28sd6jU1@uni-berlin.de>,
â ¡[â º Ð oâ «h â ]â ¡ <nomail@noaddress.not> wrote:
>We now that is possible to define the base $a>1$ logarithmic function as
>the only function f:(0,+inf) --> R such that:

>1. x<y => f(x)<f(y)
>2. f(a) = 1
>3. f(xy) = f(x)+f(y)

>Now using only this theorem (thinks as nobody knows the proof), is it
>possible to prove the existence and uniqueness of the restriction of the
>log_a to [1,+inf). i.e. that exist and is unique the function f:[1,+inf)
>--> R such that 1. 2. and 3. are satisfied?

I don't know what "thinks as nobody knows the proof" means.

You seem to be asking, if there is a unique f on (0,infty) satisfying
1,2 and 3, is there a unique f on [1,infty) satisfying the same
conditions (but for x,y in [1,infty))?

Hint: Given any such function f on [1,infty), define g on (0,infty) by
g(x) = f(x) for x >= 1, g(x) = -f(1/x) for 0 < x < 1. Prove that
g satisfies 1, 2 and 3...

Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada