Logarithmic function as an integral

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Homework Statement


I'm attempting to prove the rules for the logarithmic function using the Integral definition: Log(x)=[1,x]∫1/t dt. I think im alright with the product rule but I'm struggling with the quotient rule: i.e. Log(a/b)=Log(a)-Log(b). I believe that I'm having trouble breaking up the Integral correctly. Any help would be appreciated!

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The Attempt at a Solution

 

Answers and Replies

  • #2
fzero
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If you can show the product rule, then the quotient rule will follow the same logic, with an appropriate assumption about which of a or a/b is larger. Alternatively, use substitution to show that [itex]\log(1/b) = - \log b[/itex] so that you can combine this with the product rule.
 

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