Definition:Natural Logarithm/Positive Real/Definition 2
Definition
Let $x \in \R$ be a real number such that $x > 0$.
The (natural) logarithm of $x$ is defined as:
- $\ln x := y \in \R: e^y = x$
where $e$ is Euler's number.
That is:
- $\ln x = \log_e x$
Notation
The notation for the natural logarithm function is misleadingly inconsistent throughout the literature. It is written variously as:
- $\ln z$
- $\log z$
- $\Log z$
- $\log_e z$
The first of these is commonly encountered, and is the preferred form on $\mathsf{Pr} \infty \mathsf{fWiki}$. However, many who consider themselves serious mathematicians believe this notation to be unsophisticated.
The second and third are ambiguous (it doesn't tell you which base it is the logarithm of).
While the fourth option is more verbose than the others, there is no confusion about exactly what is meant.
Also see
- Results about logarithms can be found here.
Historical Note
The natural logarithm was discovered by accident by John Napier in around $1590$, evolving from his invention of the Napierian logarithm as a tool for multiplication of numbers by addition.
He had no concept of the notion of the base of a logarithm and certainly did not use Euler's number $e$.
Linguistic Note
The word logarithm comes from the Ancient Greek λόγος (lógos), meaning word or reason, and ἀριθμός (arithmós), meaning number.
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.6$. The Logarithm
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Natural 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: $(15)$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): logarithmic function
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): logarithmic function