Definition:Natural Logarithm/Positive Real/Definition 3

Definition

Let $x \in \R$ be a real number such that $x > 0$.

The (natural) logarithm of $x$ is defined as:

$\ds \ln x := \lim_{n \mathop \to \infty} n \paren {\sqrt [n] x - 1}$

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

• Equivalence of Definitions of Real Natural Logarithm

• 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

• 1908: G.H. Hardy: A Course of Pure Mathematics: $\S 68$