Question

What is the domain of the natural logarithm?

Answer 1

See explanation...

Explanation:

A natural logarithm is the inverse of the Real exponential function `x -> e^x` or the Complex exponential function `z -> e^z`.

The domain of `e^x` is the whole of `RR`, while its range is `(0, oo)`. The exponential function is one-one on its domain, so its inverse - the Real natural logarithm, `ln x` - is well defined, with domain `(0, oo)` and range `RR`.

The Complex logarithm is somewhat messier. `z -> e^z` is many to one since `e^(2pii) = 1`. So we need to restrict the domain of `e^z` or the range of `ln z` in order to be able to define `ln z` as a function.

We can define the principal natural logarithm of a Complex number as follows:

`ln z = ln abs(z) + "Arg"(z) i`

The definition of (the principal value of) `"Arg"(z)` may be in the range `(-pi, pi]` or `[0, 2pi)`, resulting in different possible definitions for the principal Complex logarithm.

The one number for which we cannot define a meaningful logarithm is `0`.