I read the domain of a logarithmic function can not be negative, but why?
I mean, a logarithmic function with base $-2$ and parameter $-1/8$ should result in the exponent $-3$ shouldn't it?
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If you're prepared for complex results, then I don't see why you can't have a negative base. I presume you know the base change formula? – J. M. ain't a mathematician May 21 '17 at 12:59
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Do you define logarithm as inverse of exponential functions? – velut luna May 21 '17 at 12:59
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@velutluna first line of wikipedia: "In mathematics, the logarithm is the inverse operation to exponentiation.", so kinda. – JonasAnon May 21 '17 at 13:00
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@J.M.isn'tamathematician so you may actually do it? – JonasAnon May 21 '17 at 13:01
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2Take a look at this – kingW3 May 21 '17 at 13:01
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Well, you need to answer my question first: do you know the base change formula for logarithms? – J. M. ain't a mathematician May 21 '17 at 13:02
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$y= log_a(x)$ is equivalent to $x= a^y$. If you allow a to be negative you may run into problems with fractional powers such as $(-2)^{1/2}$. There may not be a problem with specific values such as the ones you give but we need the logarithm to be a function that applies to all positive numbers. – user247327 May 21 '17 at 13:02
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Then it depends on whether you can define $a^x$ when $a<0$. – velut luna May 21 '17 at 13:03
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@J.M.isn'tamathematician yes, I do, and I see what goes wrong there, but it shouldn't take away the idea behind the function, because the example i gave should work out. – JonasAnon May 21 '17 at 13:04
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"I see what goes wrong there" - then you know that it's possible to choose a branch of the complex logarithm apart from the principal branch, no? – J. M. ain't a mathematician May 21 '17 at 13:06
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if you mean logical thinking, yes – JonasAnon May 21 '17 at 13:12
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Well.. logarithms are functions, functions with some properties. The basic rule of defining a function is that it should be useful.
You may argue with your examples like $(-2)^3=-8$ but I can counter with things like $(-2)^n=4$. You surely don't want to make changes in a function which only apply for odd powers of base.
Furthermore, when you make changes and start dealing with things like log$_{a} x$ when x is negative then you get multiple results, or your logarithmic function becomes multi valued. For instance $(-2)^3=-8$ but $(-2)^{1.09284065−0.420787248i}=-8$ too. Remember that multi valued function thing? I hope its enough.
Vidyanshu Mishra
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