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February 20th, 2008, 12:15 PM  #1 
Newbie Joined: Feb 2008 Posts: 16 Thanks: 0  help me pleae with tis question!!!
Let f(x) be defined for all x > 0 and assume the following are true: (i) f(ab) = f(a) + f(b) for all a > 0, b > 0, (ii) f'(1) = 3. (a) Show that for all x > 0, f'(x) exists and calculate f'(x). (b)Find f(x). 
February 20th, 2008, 12:36 PM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: help me pleae with tis question!!!
f(1 * 1) = f(1) + f(1) = 2f(1), so f(1) = 0. The function is essentially logarithmic.

February 24th, 2008, 01:22 AM  #3  
Newbie Joined: Feb 2008 Posts: 1 Thanks: 0  Re: help me pleae with tis question!!! Quote:
then what happens? I don't think this will help us to solve this question..  
February 24th, 2008, 01:51 PM  #4 
Senior Member Joined: Apr 2007 Posts: 2,140 Thanks: 0 
∫ln(x)dx = xln(x)  x + C Might wanna change your name, just an reminder. 
February 25th, 2008, 08:29 AM  #5 
Member Joined: Feb 2008 From: Dayton, OH, USA Posts: 33 Thanks: 0 
It is clear that f(x)=3*ln(x) however I am not sure if this solution is unique and furthermore I arrived at it only through intuition and, try as I might, I could not think of a rigorous way to show (a) or find (b).

February 25th, 2008, 10:23 AM  #6 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
johnny already told you how to find b.

February 25th, 2008, 10:32 AM  #7 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
As for uniqueness, you need only prove that one point has a unique value. Once you have that, the equation f(ab) = f(a) + f(b) shows uniqueness, since you can use that to find the value of any point based on the value of a given point.
