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

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SUMMARY

The natural logarithm function ln(-x) is defined for negative real numbers x, specifically when x < 0, as it translates to ln(-x) = ln(x) + iπ. This conclusion is supported by Euler's formula, which demonstrates the relationship between exponential and logarithmic functions in the complex plane. However, the extension of the natural logarithm to negative reals introduces complexities due to its multivalued nature, as ln(-x) can yield multiple valid results, such as ln(3) + iπ and ln(3) + 3iπ.

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

- 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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