Can Logarithms Be Defined for Negative Numbers in Advanced Mathematics?

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physicsdreams
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I recently had a test (precalc) where we had to solve log(x)-log(x+4)=2 for x.

The answer comes out negative.
I understand that in precalc we are defining the logarithms for just positive numbers, but-

Is it ever justified to define a logarithm for all numbers, both negative and positive?
(higher levels of math?)

Thanks
 

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well in calculus while going over series, my professor introduced us to euler's famous identity/formula e^(ipi)+1=0, which is a pretty cool thing, you should look it up on google if you've never seen it
anyways my professor then went on that with this formula we are able to evaluate the natural logarithm at negative values
for example e^(ipi)=-1 taking the natural log of both sides you get ipi=ln(-1)
and you can do this for other negative values as well
ln(-5)=ipi+ln5
since ln(-5)=ln(-1)+ln(5)
if you'd like to learn more, your best bet would be complex analysis i believe
 
Hi physicsdreams! :smile:

Just like the square root of a negative number is defined for complex numbers, the logarithm of negative numbers, or rather of complex numbers in general, is defined.

You can find some info and pictures here:
http://mathworld.wolfram.com/NaturalLogarithm.html
Wikipedia also has a good article, but that may be more than you're bargaining for.

However, this is rather tricky, since the logarithm for complex numbers is not a normal function any more - it is a multivalued function.
In particular this means that the rules for logarithms that you're familiar with, do not work anymore.