Definition:General Logarithm
Definition
Positive Real Numbers
Let $x \in \R_{>0}$ be a strictly positive real number.
Let $a \in \R_{>0}$ be a strictly positive real number such that $a \ne 1$.
The logarithm to the base $a$ of $x$ is defined as:
- $\log_a x := y \in \R: a^y = x$
where $a^y = e^{y \ln a}$ as defined in Powers of Real Numbers.
Complex Numbers
Let $z \in \C_{\ne 0}$ be a non-zero complex number.
Let $a \in \R_{>0}$ be a strictly positive real number such that $a \ne 1$.
The logarithm to the base $a$ of $z$ is defined as:
- $\log_a z := \set {y \in \C: a^y = z}$
where $a^y = e^{y \ln a}$ as defined in Powers of Complex Numbers.
Base of Logarithm
Let $\log_a$ denote the logarithm function on whatever domain: $\R$ or $\C$.
The constant $a$ is known as the base of the logarithm.
Examples
General Logarithm: $\log_\pi \pi$
The logarithm base $\pi$ of $\pi$ is:
- $\log_\pi \pi = 1$
General Logarithm: $\log_b 1$
- $\log_b 1 = 0$
General Logarithm: $\log_b \left({-1}\right)$
- $\log_b \left({-1}\right)$ is undefined in the real number line.
Also see
- Results about logarithms can be found here.
Linguistic Note
The word logarithm comes from the Ancient Greek λόγος (lógos), meaning word or reason, and ἀριθμός (arithmós), meaning number.