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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

    disregardthat

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    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

    disregardthat

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    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

    uart

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    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]

    whereas

    [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

    Mark44

    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.
     
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