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The natural log function

  1. Feb 15, 2013 #1
    I'm reading back over a calculus book getting ready for an exam and I'm seeing a note that I don't understand.

    It says to make sure, when rewriting a ln function that the domain is the same, then it provides an example of when it's not the same, yet says nothing more. Is this rewritten form valid?

    https://dl.dropbox.com/u/15809883/ln.png [Broken]
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Feb 15, 2013 #2


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    Staff: Mentor

    ln(x^2) = 2 ln(|x|)

    As sqrt(x^2)=|x|
  4. Feb 15, 2013 #3
    Ah yes, I had forgotten about that, thank you.
  5. Feb 15, 2013 #4


    Staff: Mentor

    The difference in the graphs is entirely due to the domains of the two different functions.

    ln(x2) is defined for all real x ≠ 0.
    2 ln(x) is defined only for x > 0.

    The rules for logarithms contain limitations on the values of the arguments. For example, ln(a*b) = ln(a) + ln(b), where a > 0 and b > 0. Note that it is possible for ln(a * b) to be defined even though the right side is undefined. This can happen when both a and b are negative.
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