Logarithm

In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is, because is to the rd power:. More generally, if, then is the logarithm of to base, written, so. As a single-variable function, the logarithm to base is the inverse of exponentiation with base.
The logarithm base is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number e | as its base; its use is widespread in mathematics and physics because of its very simple derivative. The binary logarithm uses base and is widely used in computer science, information theory, music theory, and photography. When the base is unambiguous from the context or irrelevant it is often omitted, and the logarithm is written.
Logarithms were introduced by John Napier in 1614 as a means of simplifying calculations. They were rapidly adopted by navigators, scientists, engineers, surveyors, and others to perform high-accuracy computations more easily. Using logarithm tables, tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition. This is possible because the logarithm of a product is the sum of the logarithms of the factors:
provided that, and are all positive and. The slide rule, also based on logarithms, allows quick calculations without tables, but at lower precision. The present-day notion of logarithms comes from Leonhard Euler, who connected them to the exponential function in the 18th century, and who also introduced the letter as the base of natural logarithms.
Logarithmic scales reduce wide-ranging quantities to smaller scopes. For example, the decibel is a unit used to express ratio as logarithms, mostly for signal power and amplitude. In chemistry, pH is a logarithmic measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and of geometric objects called fractals. They help to describe frequency ratios of musical intervals, appear in formulas counting prime numbers or approximating factorials, inform some models in psychophysics, and can aid in forensic accounting.
The concept of logarithm as the inverse of exponentiation extends to other mathematical structures as well. However, in general settings, the logarithm tends to be a multi-valued function. For example, the complex logarithm is the multi-valued inverse of the complex exponential function. Similarly, the discrete logarithm is the multi-valued inverse of the exponential function in finite groups; it has uses in public-key cryptography.

Motivation

, multiplication, and exponentiation are three of the most fundamental arithmetic operations. The inverse of addition is subtraction, and the inverse of multiplication is division. Similarly, a logarithm is the inverse operation of exponentiation. Exponentiation is when a number, the base, is raised to a certain power, the exponent, to give a value ; this is denoted
For example, raising to the power of gives :
The logarithm of base is the inverse operation, that provides the output from the input. That is, is equivalent to if is a positive real number.
One of the main historical motivations of introducing logarithms is the formula
by which tables of logarithms allow multiplication and division to be reduced to addition and subtraction, a great aid to calculations before the invention of computers.

Definition

Given a positive real number such that, the logarithm of a positive real number with respect to base is the exponent by which must be raised to yield. In other words, the logarithm of to base is the unique real number such that.
The logarithm is denoted "".
An equivalent and more succinct definition is that the function is the inverse function to the function.

Examples

, since.
Logarithms can also be negative: since
is approximately 2.176, which lies between 2 and 3, just as 150 lies between and.
For any base , and, since and, respectively.
Logarithmic identities

Several important formulas, sometimes called logarithmic identities or logarithmic laws, relate logarithms to one another.

Product, quotient, power, and root

The logarithm of a product is the sum of the logarithms of the numbers being multiplied; the logarithm of the ratio of two numbers is the difference of the logarithms. The logarithm of the -th power of a number is times the logarithm of the number itself; the logarithm of a -th root is the logarithm of the number divided by. The following table lists these identities with examples. Each of the identities can be derived after substitution of the logarithm definitions or in the left hand sides. In the following formulas, and are positive real numbers and is an integer greater than 1.

Identity	Formula	Example
Product
Quotient
Power
Root

Change of base

The logarithm can be computed from the logarithms of and with respect to an arbitrary base using the following formula:
Typical scientific calculators calculate the logarithms to bases 10 and. Logarithms with respect to any base can be determined using either of these two logarithms by the previous formula:
Given a number and its logarithm to an unknown base , the base is given by:
which can be seen from taking the defining equation to the power of

Particular bases

Among all choices for the base, three are particularly common. These are, . In mathematical analysis, the logarithm base is widespread because of analytical properties explained below. On the other hand, logarithms are easy to use for manual calculations in the decimal number system:
Thus, is related to the number of decimal digits of a positive integer : The number of digits is the smallest integer strictly bigger than
For example, is approximately 3.78 . The next integer above it is 4, which is the number of digits of 5986. Both the natural logarithm and the binary logarithm are used in information theory, corresponding to the use of nats or bits as the fundamental units of information, respectively.
Binary logarithms are also used in computer science, where the binary system is ubiquitous; in music theory, where a pitch ratio of two is ubiquitous and the number of cents between any two pitches is a scaled version of the binary logarithm, or log 2 times 1200, of the pitch ratio, or equivalently the log base and in photography, where rescaled base 2 logarithms are used to measure exposure values, light levels, exposure times, lens apertures, and film speeds in "stops".
The abbreviation is often used when the intended base can be inferred based on the context or discipline, or when the base is indeterminate or immaterial. Common logarithms, historically used in logarithm tables and slide rules, are a basic tool for measurement and computation in many areas of science and engineering; in these contexts still often means the base ten logarithm. In mathematics usually refers to the natural logarithm.
In computer science and information theory, often refers to binary logarithms. The following table lists common notations for logarithms to these bases. The "ISO notation" column lists designations suggested by the International Organization for Standardization.

Base	Name for log_b x	ISO notation	Other notations
2	binary logarithm		,,,
	natural logarithm		,
10	common logarithm		,
	logarithm to base

History

The history of logarithms in seventeenth-century Europe saw the discovery of a new function that extended the realm of analysis beyond the scope of algebraic methods. The method of logarithms was publicly propounded by John Napier in 1614, in a book titled Mirifici Logarithmorum Canonis Descriptio. Prior to Napier's invention, there had been other techniques of similar scopes, such as the prosthaphaeresis or the use of tables of progressions, extensively developed by Jost Bürgi around 1600. Napier coined the term for logarithm in Middle Latin, logarithmus, literally meaning, derived from the Greek + .
The common logarithm of a number is the index of that power of ten which equals the number. Speaking of a number as requiring so many figures is a rough allusion to common logarithm, and was referred to by Archimedes as the "order of a number". The first real logarithms were heuristic methods to turn multiplication into addition, thus facilitating rapid computation. Some of these methods used tables derived from trigonometric identities. Such methods are called prosthaphaeresis.
Invention of the function now known as the natural logarithm began as an attempt to perform a quadrature of a rectangular hyperbola by Grégoire de Saint-Vincent, a Belgian Jesuit residing in Prague. Archimedes had written The Quadrature of the Parabola in the third century BC, but a quadrature for the hyperbola eluded all efforts until Saint-Vincent published his results in 1647. The relation that the logarithm provides between a geometric progression in its argument and an arithmetic progression of values, prompted A. A. de Sarasa to make the connection of Saint-Vincent's quadrature and the tradition of logarithms in prosthaphaeresis, leading to the term "hyperbolic logarithm", a synonym for natural logarithm. Soon the new function was appreciated by Christiaan Huygens, and James Gregory. The notation was adopted by Gottfried Wilhelm Leibniz in 1675, and the next year he connected it to the integral
Before Euler developed his modern conception of complex natural logarithms, Roger Cotes had a nearly equivalent result when he showed in 1714 that