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

[Kamataat]

Is it true that that ln(-x) is defined for x \in R such that x < 0?

- Kamataat

 

[Muzza]

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.

 

[dextercioby]

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

Daniel.

 

[Kamataat]

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