1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Using implicit differentiation to differentiate log_a (x)

  1. Jun 22, 2006 #1
    Could someone please make sure I'm doing this right.
    I want to find the derivative of the logarithm to the base a of x, using implicit differentiation.

    Let [tex]y = \log_{a} x[/tex]

    [tex]a^y = x[/tex]

    [tex]\frac{d}{dx} (a^y) = 1[/tex] (implicit differentiation)

    [tex]\frac{d}{dx} (e^{\ln a})^y = 1[/tex]

    [tex]\frac{d}{dx} (e^{(\ln a)y}) = 1[/tex]

    [tex]e^{(\ln a)y} \frac{d}{dx} ((\ln a)(y)) = 1[/tex]

    [tex](e^{\ln a})^y \frac{d}{dy} ((\ln a)(y)) \frac{dy}{dx} = 1[/tex] (did I use the chain rule correctly here?)

    [tex](a^y)(\ln a) \frac{dy}{dx} = 1[/tex]

    [tex]x \ln a \frac{dy}{dx} = 1[/tex]

    [tex]\frac{dy}{dx} = \frac{1}{x \ln a}[/tex]
    Last edited: Jun 22, 2006
  2. jcsd
  3. Jun 22, 2006 #2


    User Avatar
    Staff Emeritus
    Science Advisor

    Yep, very nicely done!
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Using implicit differentiation to differentiate log_a (x)