Given a logarithm is true, if and only if, y=logbx and by=x (and x and b are positive, and b is not equal to 1)[1], are true, why aren't logarithms defined for negative numbers?
Why can't b be negative? Take (−2)3=−8 for example. Turning that into a logarithm, we get 3=log(−2)(−8) which is an invalid equation as b and x are not positive! Why is this?
Answer
In complex analysis, x can be negative. For example eiπ=−1, so ln(−1)=iπ.
I hadn't seen a log with a negative base, but I thought one could define it with the normal change of base formula: logbx=lnxlnb. However, this turns out to be inconsistent Might be inconsistent, at the very least, it doesn't give 3:
log(−2)(−8)=ln(−8)ln(−2)=ln8+iπln2+iπ=3ln2+3iπ−2iπln2+iπ=3−2iπln2+iπ≠3
This is because the complex log has a branch cut in it. For example: e3iπ=−1, but ln(−1)=iπ, not 3iπ. The cut is made so that Im(lnz)∈(−π,π].
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