Natural logarithm

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The natural logarithm is the logarithm to the base e, where e is an irrational 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).

The natural logarithm of a number x (written as ln(x)) is the power to which e would have to be raised to equal x. For example, ln(7.389...) is 2, because e2=7.389.... The natural log of e itself (ln(e)) is 1 because e1 = e, while the natural logarithm of 1 (ln(1)) is 0, since e0 = 1.

The natural logarithm can be defined for all positive real numbers 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:

Like all logarithms, the natural logarithm maps multiplication into addition:

Thus, the logarithm function is an isomorphism from the group of positive real numbers under multiplication to the group of real numbers under addition, represented as a function:

Logarithms can be defined to any positive base other than 1, not just e; however logarithms in other bases differ only by a constant multiplier from the natural logarithm, and are usually defined in terms of the latter. Logarithms are useful for solving equations in which the unknown appears as the exponent of some other quantity. For example, logarithms are used to solve for the half-life, decay constant, or unknown time in exponential decay problems. They are important in many branches of mathematics and the sciences and are used in finance to solve problems involving compound interest.

  Part of a series of articles on
The mathematical constant e

Natural logarithm · Exponential function

Applications in: compound interest · Euler's identity & Euler's formula  · half-lives & exponential growth/decay

Defining e: proof that e is irrational  · representations of e · Lindemann–Weierstrass theorem

People John Napier  · Leonhard Euler

Schanuel's conjecture

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