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(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 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:

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.


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

Read more about Natural Logarithm:  History, Notational Conventions, Origin of The Term natural Logarithm, Definitions, Properties, Derivative, Taylor Series, The Natural Logarithm in Integration, Numerical Value, Continued Fractions, Complex Logarithms

Other articles related to "logarithms, logarithm, natural logarithm":

Logrithm - History - From Napier To Euler
... The method of logarithms was publicly propounded by John Napier in 1614, in a book entitled Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Rule of ... Joost Bürgi independently invented logarithms but published six years after Napier ... Johannes Kepler, who used logarithm tables extensively to compile his Ephemeris and therefore dedicated it to Napier, remarked...the accent in calculation led Justus ...
Logrithm - Analytic Properties - Integral Representation of The Natural Logarithm
... The natural logarithm of t agrees with the integral of 1/x dx from 1 to t In other words, ln(t) equals the area between the x axis and the graph of the function 1/x, ranging ... side of this equation can serve as a definition of the natural logarithm ... Product and power logarithm formulas can be derived from this definition ...
e (mathematical Constant)
... approximately equal to 2.71828, that is the base of the natural logarithm ... the exponential function, and its inverse is the natural logarithm, or logarithm to base e ... The natural logarithm of a positive number k can also be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case, e is the ...
Natural Logarithm - Complex Logarithms
... This exponential function can be inverted to form a complex logarithm that exhibits most of the properties of the ordinary logarithm ... So the logarithm cannot be defined for the whole complex plane, and even then it is multi-valued – any complex logarithm can be changed into an "equivalent ... The complex logarithm can only be single-valued on the cut plane ...
Partial Function - Discussion and Examples - Natural Logarithm
... Consider the natural logarithm function mapping the real numbers to themselves ... The logarithm of a non-positive real is not a real number, so the natural logarithm function doesn't associate any real number in the codomain with any non-positive real number in the domain ... Therefore, the natural logarithm function is not a total function when viewed as a function from the reals to themselves, but it is a partial function ...

Famous quotes containing the word natural:

    Numerous studies have shown that those adults who feel the most frustrated by children—and the least competent as parents—usually have one thing in common.... They don’t know what behaviors are normal and appropriate for children at different stages of development. This leads them to misinterpret their children’s natural behaviors and to have inappropriate expectations, both for their children and themselves.
    Lawrence Kutner (20th century)