# Logarithmic function defined as an integral

• JMR_2413
In summary, the person is attempting to prove the rules for the logarithmic function using the integral definition. They are having trouble with the quotient rule and are seeking help. The suggestion is to use the Chain Rule to find all the properties of the log.

#### 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!

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.

JMR_2413 said:
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.