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

Oct 31, 2015

Yes and yes, but it gets complicated...

#### Explanation:

${e}^{x} : \mathbb{R} \to \left(0 , \infty\right)$ is a one to one Real-valued function with inverse $\ln : \left(0 , \infty\right) \to \mathbb{R}$

If $0 < x < 1$ then $\ln x < 0$

${e}^{z} : \mathbb{C} \to \mathbb{C} \setminus \setminus \left\{0\right\}$ 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,

${e}^{z} : \left\{a + i b \in \mathbb{C} : - \pi < b \le \pi\right\} \to \mathbb{C} \setminus \setminus \left\{0\right\}$ is a one one function.

If we use this definition then

$\ln z : \mathbb{C} \setminus \setminus \left\{0\right\} \to \left\{a + i b \in \mathbb{C} : - \pi < b \le \pi\right\}$

is well defined and we find values like $\ln \left(- 1\right) = i \pi$