Logarithmic function as an integral

[JMR_2413]

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 I am 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!

Homework Equations

The Attempt at a Solution

 

[fzero]

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 \log(1/b) = - \log b so that you can combine this with the product rule.

 

[HallsofIvy]

This is pretty much the same question as
https://www.physicsforums.com/showthread.php?t=551610