Logarithm

For a non-zero complex number $z$, $\log z$ is the set of all $w \in \mathbb{C}$ with $e^w = z$.

Writing $z = re^{i\theta}$ with $r = |z|$ and $w = u + iv$, the condition $e^w = z$ gives $e^u = r$ and $v \in \arg z$. So

Here:

• $\operatorname{Arg} z$: principal argument of $z$, the unique argument with $-\pi < \operatorname{Arg} z \le \pi$
• $\arg z = \{\operatorname{Arg} z + 2k\pi : k \in \mathbb{Z}\}$

$\log z$ is multi-valued.

Principal Value

The logarithm value deduced by setting $k=0$.

Below you can change $z$ and see how $\operatorname{Log}(z)$ changes.

Properties

Let $z_1$ and $z_2$ be non-zero.

• $\log(z_1 z_2) = \log z_1 + \log z_2$
• $\log\dfrac{z_1}{z_2} = \log z_1 - \log z_2$
• $\log\dfrac{1}{z} = -\log z$

Examples

$\log(1+i)$

• $|1+i| = \sqrt{2}$
• $\operatorname{Arg}(1+i) = \dfrac{\pi}{4}$

$\log i$

• $|i| = 1$
• $\operatorname{Arg}(i) = \dfrac{\pi}{2}$