Monday, 22 February 2016

number theory - Why aren't logarithms defined for negative x?



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+3iπ2iπln2+iπ=32iπln2+iπ3



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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