# Logarithmic function defined as an integral

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!

AlephZero
Homework Helper
If you have already proved Log(a) + Log(b) = Log(ab) from the integral definition
and you want to prove Log(a/b) + Log(b) = Log(a),
just replace a by a/b everywhere in your proof.

lavinia