Logarithmic function defined as an integral

[JMR_2413]

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!

 

[AlephZero]

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]

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!

Try using the Chain Rule.

d/dxlog(a/x) = (x/a)(-a/x^2) = -1/x so log(a/x) = -log(x) + c.

Now calculate c.

The Chain Rule can be used to find all of the properties of the log once one sets log(1) to equal zero.