# Definitions for **natural logarithm**

### This page provides all possible meanings and translations of the word **natural logarithm**

### Random House Webster's College Dictionary

nat′ural log′arithm(n.)

a logarithm having e as a base.

**Category:**Math**Ref:**Symbol: ln; Also called Napierian logarithm.; Compare common logarithm.

### Princeton's WordNet

natural logarithm, Napierian logarithm(noun)

a logarithm to the base e

### Wiktionary

natural logarithm(Noun)

The logarithm in base e; either the function that given returns such that or the value of .

**Origin:** From log naturalis, coined in 1668 by Nicholas Mercator in his book Logarithmo-technica.

### Freebase

Natural logarithm

The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828. The natural logarithm is generally written as ln x, loge x or sometimes, if the base of e is implicit, as simply log x. Parentheses are sometimes added for clarity, giving ln(x), loge(x) or log(x). This is done in particular when the argument to the logarithm is not a single symbol, in order to prevent ambiguity. The natural logarithm of a number x is the power to which e would have to be raised to equal x. For example, ln is 2, because e²=7.389.... The natural log of e itself is 1 because e¹ = e, while the natural logarithm of 1 is 0, since e0 = 1. The natural logarithm can be defined for any positive real number a as the area under the curve y = 1/x from 1 to a. The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, leads to the term "natural." The definition can be extended to non-zero complex numbers, as explained below. The natural logarithm function, if considered as a real-valued function of a real variable, is the inverse function of the exponential function, leading to the identities:

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