# Logarithm

In Mathematical, a function logarithm is a function $f$ definite on $\right] 0, + \ infty with values in\ R, \left[\left[continuity|continuous\right] and processing a product all in all, i.e checking$

$\ forall has \ in\right] 0, + \ infty \ forall B \ in 0, + \ infty F \left(ab\right) =f \left(a\right)+f \left(b\right)$

This property imposes that any function logarithm is null into 1.

Property stated above rises that any function logarithm is necessarily a bijection of $\right] 0, + \ infty on\ R, and the reciprocal image of 1 by this function is called the \text{'}\text{'} bases \text{'}\text{'} logarithm. Reciprocally, B a real number strictly positive and different from 1 is$. There then exists a single function logarithm being worth 1 in B . One calls this function the basic logarithm B , noted $\ log_b \left(X\right)$, and it is the function which with X associates the power for which it is necessary to raise B to find X , C. - with-D. $b^ \left\{\ log_b \left(X\right)\right\} = x$. The functions logarithms are thus the Réciproque S of the exponential functions.

The most known functions logarithms are the natural or Napierian Logarithme basic $e$, the Decimal logarithm (basic 10, very much used in physics) and the binary logarithm (basic 2, used in data processing, in particular in theory of complexity). The logarithms thereafter were generalized in the complex plan (complex logarithms) by analytical prolongation and were introduced in theory of the groups (discrete logarithms) by analogy with the analysis.

## History of the logarithms

In 1588, to facilitate his calculations, the Swiss astronomer Jost Bürgi developed the first known system logarithmic curve.

When in 1614, John Napier or Neper its treaty Mirifici Logarithmorum Canonis Descriptio publishes, he does not think that he is creating new functions, but only tables of correspondences (logos = report/ratio, relation, arithmeticos = number) between two series of values having the following property: to a product in a column, a sum in another corresponds. These tables of correspondences were created initially to simplify trigonometrical calculations appearing in astronomical calculations and will be used a few years later by Kepler. In 1619, a posthumous work of Neper Mirifici Logarithmorum Canonis Constructio appears, where he explains how to build a table logarithmic curve (see the article Table of logarithms to include/understand the principle of it). Its work will be continued and prolonged by the English mathematician Henry Briggs which publishes the decimal tables of logarithms and specifies the methods of use of the tables to calculate sines, to find angles of tangents… The decimal logarithm is sometimes called logarithm of Briggs in its honor.

In 1647, when Gregoire of Saint-Vincent works on the squaring of the hyperbole, it highlights a new function which is being the primitive of the function $x \ to 1/x$ cancelling itself into 1 but it is only Huygens into 1661 which will notice that this function is being a particular function logarithm: the natural Logarithm.

The concept of function, the correspondence between the exponential functions and the functions logarithms only appear more tardily after the work of Leibniz on the concept of function (1697).

## Decimal logarithm

It is the most practical logarithm in numerical calculations, it is noted log or $\ log_ \left\{10\right\}$. One finds it in the creation of the logarithmic scales, the semi-logarithmic reference marks or log-log, in the Rule slide, in the calculation of pH, in the unit of the Décibel.

It specifies to which power of 10 a number corresponds:

log (10) = 1, log (100) = 2, log (1000) = 3, log (0,01) = - 2.

Remain that the value of the logarithm of other numbers that powers of 10 request an approximate calculation. The calculation of log (2) for example can be done with the hand, by noticing that $2^ \left\{10\right\} \ approx 1000$ thus $10 \ log \left(2\right) \ approx 3$ thus $\ log \left(2\right) \ approx 0,3$

## Natural logarithm

It is the logarithm whose Dérivée is simplest. It is the fact besides that it is a Primitive of $x \ to 1/x$ which gave him this importance. It is noted Log or ln . On the other hand, when it was necessary to seek the base of this logarithm, the mathematicians did not fall on a very simple value: the base of this logarithm is a number, neither decimal, neither rational, nor algebraic: it is the transcendent number E $\ approx 2,718 281 \ cdots$.

Note: this logarithm is also named Napierian logarithm

## Properties of the functions logarithms

For all real has strictly positive and different from 1, the basic logarithm has : $\ log_a \,$ is the function continues definite on $\right] 0; + \ infty checking, for all \text{'}\text{'} X \text{'}\text{'} and \text{'}\text{'} there \text{'}\text{'} real strictly positive :\ log_a \left(xy\right) = \ log_a \left(X\right) + \ log_a \left(there\right) \,andlog_a \left(a\right) = 1 \,This definition makes it possible to deduce the following properties quickly$
$\ log_a \left(1\right) = 0 \,$
$\ log_a \left(x/y\right) = \ log_a \left(X\right) - \ log_a \left(there\right) \,$
$\ log_a \left(x^n\right) =n \ log_a \left(X\right) \,$
$\ log_a \left(a^n\right) = N \,$ for entire naturalness N , then for entire relative N
$\ log_a \left(a^r\right) = R \,$ for very rational R .
As any reality X can be regarded as limit of terms of the form $a^ \left\{r_n\right\} \,$, one determines $log_a \left(X\right) \,$ like the limit of $r_n \,$.

Two functions logarithms differ only from one multiplicative constant near:

$\ log_b \left(X\right) = \ frac \left\{\ log_a \left(X\right)\right\}\left\{\ log_a \left(b\right)\right\}$
$\ log_a \left(b\right) \ log_b \left(X\right) = \ log_a \left(X\right) \,$
Indeed $log_b$ is the function continues which all in all processes a product and which is worth 1 in B , but the function $\ frac \left\{\ log_a\right\} \left\{\ log_a \left(b\right)\right\}$ is also a continuous function which all in all processes a product and which is worth 1 out of B. The two functions are thus identical.

All the functions logarithms can thus be expressed using one only, one which one knows already the derivative: ln

$\ log_a \left(X\right) = \ frac \left\{\ ln \left(X\right)\right\}\left\{\ ln \left(a\right)\right\}$

The function $log_a$ is derivable of derived:

$\ log_a\text{'} \left(X\right) = \ frac \left\{1\right\} \left\{X \ ln \left(a\right)\right\}$
It is thus strictly monotonous, increasing when has is higher than 1, decreasing in the contrary case.

It is a bijection whose reciprocal one is the function $x \ to a^x$

mathematical Curiosity: with an error lower than 0,6%

$\ log_2 \left(X\right) \ approx \ log_ \left\{10\right\} \left(X\right) + \ ln \left(X\right) \,$.

## See too

### Related articles

• natural Logarithme
• Decimal logarithm
• Logarithme complexes
• holomorphic Fonction
• Exponentielle
• the presentation of the logarithms in Wikilivre of photography Photographie - Chapter 01 - a Little mathematics

### External bonds

Simple: Logarithm

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