Here is a very simple problem which appeared with my first year calculus students. Consider the function

Its domain is . But now, we may take the exponent down, using the rules of the logarithm…

and now, the domain is … What happens??? Try to explain it: (a) *without* complex numbers, (b) *with* them…

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Oh, now i understand with a nephritic spasm is called

calculusCan this be done counting with fingers?

It depends on how many fingers you have. If the answer is complex, then you can XD

Does it have something to do with the fact that log(-1))=i pi? :)

You bet… XD

I don´t know exacly what happens but i used octave to plot both graphs and they are the same, the have the same domain R-{0}.

So I try to find a explanation, but wikipedia doesn´t help very much xD.

Octave does funny things, you have to be careful. Try log(-1), and you get a complex number (0.000 + 3.14159 i = i pi). Yet, when you plot it, it only plots the modulus! So, if you try to plot log(x), it is as if its domain was the whole of R and the function is even. So, don’t rely much on octave for these things…

I can’t imagine any explanation with real numbers only, so I’ll give it with complex.

I start with exp(i pi)=-1, so log(-1)=i pi. Now, let us consider a negative x, so x=-|x|. Then, log(x)=log(-1*|x|) = log(-1) + log(|x|) = i pi + log(|x|).

So, 2 log(x) = 2pi i + 2 log(|x|), if x<0

But 2pi i is the log of 1! So you have 2 log(x) = log(1) + 2 log(|x|) = 2 log(|x|)

Of course, you've changed sheet in the Riemann surface, but who cares? :)

Anyway, I feel the explanation is too difficult… I’m striving for ideas!

The best explanation is yours (maybe it can be written in a more elegant way). We need to think that log (x) has a discontinuity in x=0, so the best way to work with is to rewrite it as a piecewise function.

That leads to your explanation.

By now I’m triying to recover a completely wrong, but funny, explanation that i’ve read long time ago in Boyer’s History of Maths, but I haven’t found it yet. I think I’ll have to read it whole again.

I haven’t use english for such a long time, so sorry about dictionary kicking ;-P

Paco, welcome!! :) Please try to fish out that funny explanation. I think I read it too, but I can’t recall it now. About your English… it’s perfectly fine. Do you think anybody here has the right to complain? ;)