Definition:General Logarithm/Positive Real
Definition
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.
The act of performing the $\log_a$ function is colloquially known as taking logs.
Common Logarithms
Logarithms base $10$ are often referred to as common logarithms.
Binary Logarithms
Logarithms base $2$ are becoming increasingly important in computer science.
They are often referred to as binary logarithms.
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.
Also known as
The logarithm to the base $a$ of $x$ is usually voiced in the abbreviated form:
- log base $a$ of $x$
or
- log $a$ of $x$.
When $a = 2$, a notation which is starting to take hold for $\log_2 x$ is $\lg x$.
This concept is becoming increasingly important in computer science.
Also see
- Definition:Real Natural Logarithm: when $a = e$
- 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.
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Logarithms and Antilogarithms
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms: $(9)$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): base: 2. (of logarithms)
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): logarithm (log)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): base: 2. (of logarithms)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): logarithm (log)
- 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 13$: Logarithms and Antilogarithms