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Logarithm question

  1. Feb 16, 2005 #1
    Is it true that that [itex]ln(-x)[/itex] is defined for [itex]x \in R[/itex] such that [itex]x < 0[/itex]?

    - Kamataat
     
  2. jcsd
  3. Feb 16, 2005 #2
    Well, yes... If x < 0 then -x > 0, and surely the natural logarithm is defined for all positive numbers.

    I believe it can be extended to the negative reals as well, by using Euler's formula. For example, e^(ln(3) + ipi) = e^ln(3) * e^(ipi) = -3, so one might say that ln(-3) = ln(3) + ipi. I assume there are some technical difficulties in actually making such an extension rigorous, since (for example) ln(3) + 3ipi is also a possible "candidate" for being the natural logarithm of -3.
     
  4. Feb 16, 2005 #3

    dextercioby

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    Complex logarithm is multivalued indeed.But that's another story.

    Daniel.
     
  5. Feb 16, 2005 #4
    I thought it was so (I mean, it's pretty obvious), but I had this weird doubt (sometimes that happens when I study maths), so had to ask. Thanks again, Muzza.

    - Kamataat
     
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