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February 20th, 2008, 01:15 PM   #1
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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).
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February 20th, 2008, 01:36 PM   #2
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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.
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February 24th, 2008, 02:22 AM   #3
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Re: help me pleae with tis question!!!

Quote:
Originally Posted by CRGreathouse
f(1 * 1) = f(1) + f(1) = 2f(1), so f(1) = 0. The function is essentially logarithmic.

then what happens? I don't think this will help us to solve this question..
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February 24th, 2008, 02:51 PM   #4
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∫ln(x)dx = xln(x) - x + C

Might wanna change your name, just an reminder.
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February 25th, 2008, 09:29 AM   #5
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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).
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February 25th, 2008, 11:23 AM   #6
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johnny already told you how to find b.
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February 25th, 2008, 11:32 AM   #7
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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.
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