Can I find the natural log of a negative number?

1 Answer
George C.
Nov 5, 2015

Yes, if x < 0 then the principal value of ln(x) is ln(-x)+i pi

Explanation:

The Real valued function e^x:RR -> (0, oo) is one to one, with inverse function ln(x):(0, oo)->RR.

We can extend the definition of e^x to the Complex valued function e^z:CC->CC\{0}, but this is a many to one function, so it has no inverse function, unless we do something to limit the domain of e^z or the range of ln z.

For example, if we limit the domain of e^z to the set {a+ib in CC : -pi < b <= pi}, then it is a one to one function with inverse function:

ln(z):CC\{0} -> {a+ib in CC : -pi < b <= pi}

If x < 0, then e^(ln(-x)+i pi) = e^ln(-x) e^(i pi) = -x * -1 = x

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