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Does a negative natural log of a negative number cancel to become a positive log?

  1. Jan 29, 2013 #1
    For instance, say I have

    -ln(-∞)​

    Does the negative sign on the natural log cancel with the negative sign on the infinity?

    Is this true?
    -ln(-∞) = ln(∞)​

    Thank you

    -Drc
     
  2. jcsd
  3. Jan 29, 2013 #2

    tiny-tim

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    Hi DrCrowbar! :smile:

    There is no such thing as the ln of a negative number. :wink:

    (unless we're allowed complex numbers, in which case eg ln(-1) = πi)
     
  4. Jan 29, 2013 #3
    Hi Tim!

    Ah... clumsy me. I knew that. I really did. Well, used to... :wink:

    So if you have ln(-∞), is that essentially ∞ or -∞?

    I'm finding the solution for a calculus III improper integrals question. I guess it doesn't matter, though, because either way it diverges (a possible solution).

    Thanks again

    -Drc
     
  5. Jan 29, 2013 #4

    Mark44

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    No, it's simply not defined, as tiny-tim said. I'm assuming you're working with real numbers.
     
  6. Jan 29, 2013 #5

    tiny-tim

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    No, that's ∞i :wink:
     
  7. Jan 29, 2013 #6
    Ah, ok.

    So ln(-#) is the same as -ln(#)... That makes sense, actually. I had forgotten what the graph of the natural log function looks like.

    Thanks guys. It's the first time I've asked a math question online and actually received a correct answer!

    -Drc
     
  8. Jan 29, 2013 #7

    tiny-tim

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    no it isn't!!

    ln(-#) is an imaginary number (something times i)

    if we're only allowed to use real numbers, then ln(-#) doesn't exist!
     
  9. Jan 29, 2013 #8
    Oh, ok.

    So you can have a negative natural log function (-ln|cscx+cotx| for instance) but you cannot have the natural log of a negative number unless you involve imaginary numbers.

    I haven't really seen much of imaginary numbers, but I hear they're used a bit in D.E.
     
    Last edited: Jan 29, 2013
  10. Jan 29, 2013 #9

    tiny-tim

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    yes, they'll be useful later

    for now, forget about them

    the graph of ln(x) is like the graph of √x …

    it simply has no value for x < 0
     
  11. Jan 29, 2013 #10

    micromass

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    What?? In what context is [itex]ln(-\infty)=\infty i[/itex]? What does [itex]infty i[/itex] even mean?
     
  12. Jan 29, 2013 #11

    tiny-tim

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    oops! :redface:

    i should have said ln(-∞) = πi + ∞ :rolleyes:
     
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