Question

Can the logarithm of a number be negative? Can it be imaginary?

Answer 1

Yes and yes, but it gets complicated...

Explanation:

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

If `0 < x < 1` then `ln x < 0`

`e^z:CC->CC \ {0}` is a many to one Complex-valued function. As a result, it has no inverse function. However, it is possible to extend the definition of logs to Complex numbers. We can limit the domain of `e^z` to make it a one to one function, allowing the definition of an inverse "`ln z`".

For example,

`e^z:{a+ib in CC : -pi < b <= pi} -> CC \ {0}` is a one one function.

If we use this definition then

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

is well defined and we find values like `ln(-1) = i pi`