Is ln(-x) Defined for Negative Real Numbers?

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  • Thread starter Thread starter Kamataat
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    Logarithm
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Discussion Overview

The discussion centers on whether the natural logarithm of negative numbers, specifically ln(-x) for x < 0, is defined within the context of real and complex analysis. The conversation explores the implications of extending the logarithm to negative reals and touches on the complexities of the complex logarithm.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • Kamataat questions if ln(-x) is defined for x < 0.
  • One participant asserts that since -x > 0 when x < 0, ln(-x) is defined for positive numbers, suggesting that it can be extended to negative reals using Euler's formula.
  • This participant proposes that ln(-3) could be expressed as ln(3) + ipi, while noting potential technical difficulties in making this extension rigorous.
  • Another participant acknowledges the multivalued nature of the complex logarithm but indicates that it is a separate topic.
  • Kamataat expresses initial doubt about the definition but appreciates the clarification provided by others.

Areas of Agreement / Disagreement

Participants generally agree that ln(-x) can be defined for x < 0 in a certain context, but there is no consensus on the rigor of extending this definition to negative reals or the implications of the multivalued nature of the complex logarithm.

Contextual Notes

The discussion highlights the dependence on definitions and the complexities involved in extending the natural logarithm to negative numbers, particularly in the context of complex analysis.

Kamataat
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Is it true that that [itex]ln(-x)[/itex] is defined for [itex]x \in R[/itex] such that [itex]x < 0[/itex]?

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

Daniel.
 
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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