Question

Can I find the natural log of a negative number?

Answer 1

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`