Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Logarithmic function defined as an integral

  1. Nov 17, 2011 #1
    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!
  2. jcsd
  3. Nov 18, 2011 #2


    User Avatar
    Science Advisor
    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.
  4. Nov 18, 2011 #3


    User Avatar
    Science Advisor
    Gold Member
    2017 Award

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook