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

Why is the deivative 2ln(x) is the same as that of ln(x^2)

  1. Aug 22, 2011 #1
    In the textbooks some times they simplify the the logarithmic using logarithmic properties . But sometimes the domain of the simplified form is not the same as that of not simplified so how their derivative are equal. ???????
  2. jcsd
  3. Aug 22, 2011 #2


    User Avatar
    Science Advisor

    The domains of the derivatives are not equal.
  4. Aug 22, 2011 #3
    I know but mustn't we simplify and take the derivative ,right ?
  5. Aug 22, 2011 #4


    User Avatar
    Science Advisor

    We don't have to, but we can. However note that 2log(x) is not a simplification of log(x^2) for negative x.
  6. Aug 22, 2011 #5
    That is what I am talking about.
  7. Aug 22, 2011 #6
    They are the same. The derivative of 2*ln(x) = 2/x.

    The derivative of ln(x^2) = (1/x^2) * 2x = 2/x.
  8. Aug 22, 2011 #7


    User Avatar
    Science Advisor

    Yep I can see that Mahmoud. :smile:

    Yes you make a good point. I sometimes see them treated as being identical when as you point out they are not.

    [tex]\frac{d}{dx} \left( 2 \log_e(x) \right) = \frac{2}{x} \,\,\,\,\,\,:\,\,\,\, x>0[/tex]


    [tex]\frac{d}{dx} \left(\log_e(x^2) \right) = \frac{2}{x} \,\,\,\,\,\,:\,\,\,\, x \neq 0[/tex]
  9. Aug 22, 2011 #8
    Isn't the point just that the original functions 2log(x) and log(x^2) have different domains? This question really has nothing to do with derivatives. But it's a good point ... you can't arbitrarily apply the log laws without double-checking the domains.
  10. Aug 22, 2011 #9
    Thanks for you all . Every time I use logarithmic laws I check the domains but my textbook doesn't mention that.
  11. Aug 22, 2011 #10


    Staff: Mentor

    If your textbook simplifies ln(x2) to 2 ln(x), they are probably making the assumption that x > 0. If they don't show that assumption anywhere, then they are being very sloppy.
  12. Aug 22, 2011 #11
    No , this problem doesn't exist but I wanted to say that why the textbook didn't mention that I must check the domains , Is the author wants me to do it myself and actually I did.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook