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Template:Arithmetic operations
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 Template:Math to base Template:Math is Template:Math, because Template:Math is Template:Math to the Template:Mathrd power: Template:Math. More generally, if Template:Math, then Template:Mvar is the logarithm of Template:Mvar to base Template:Mvar, written Template:Math, so Template:Math. As a single-variable function, the logarithm to base Template:Mvar is the inverse of exponentiation with base Template:Mvar.
The logarithm base Template:Math is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number [[e (mathematical constant)|Template:Math]] as its base; its use is widespread in mathematics and physics because of its very simple derivative. The binary logarithm uses base Template:Math 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 Template:Math.
Logarithms were introduced by John Napier in 1614 as a means of simplifying calculations.<ref>Template:Citation</ref> 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: <math display="block"> \log_b(xy) = \log_b x + \log_b y,</math> provided that Template:Mvar, Template:Mvar and Template:Mvar are all positive and Template:Math. 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 Template:Mvar as the base of natural logarithms.<ref>Template:Citation</ref>
Logarithmic scales reduce wide-ranging quantities to smaller scopes. For example, the decibel (dB) is a unit used to express ratio as logarithms, mostly for signal power and amplitude (of which sound pressure is a common example). 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.
MotivationEdit
Addition, 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 Template:Mvar, the base, is raised to a certain power Template:Mvar, the exponent, to give a value Template:Mvar; this is denoted <math display="block">b^y=x.</math> For example, raising Template:Math to the power of Template:Math gives Template:Math: <math>2^3 = 8.</math>
The logarithm of base Template:Mvar is the inverse operation, that provides the output Template:Mvar from the input Template:Mvar. That is, <math>y = \log_b x</math> is equivalent to <math>x=b^y</math> if Template:Mvar is a positive real number. (If Template:Mvar is not a positive real number, both exponentiation and logarithm can be defined but may take several values, which makes definitions much more complicated.)
One of the main historical motivations of introducing logarithms is the formula <math display="block">\log_b(xy)=\log_b x + \log_b y,</math> 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.
DefinitionEdit
Given a positive real number Template:Mvar such that Template:Math, the logarithm of a positive real number Template:Mvar with respect to base Template:MvarTemplate:Refn is the exponent by which Template:Mvar must be raised to yield Template:Mvar. In other words, the logarithm of Template:Mvar to base Template:Mvar is the unique real number Template:Mvar such that <math>b^y = x</math>.<ref>Template:Citation, chapter 1</ref>
The logarithm is denoted "Template:Math" (pronounced as "the logarithm of Template:Mvar to base Template:Mvar", "the Template:Nowrap logarithm of Template:Mvar", or most commonly "the log, base Template:Mvar, of Template:Mvar").
An equivalent and more succinct definition is that the function Template:Math is the inverse function to the function <math>x\mapsto b^x</math>.
ExamplesEdit
- Template:Math, since Template:Math.
- Logarithms can also be negative: <math display="inline">\log_2 \! \frac{1}{2} = -1</math> since <math display="inline">2^{-1} = \frac{1}{2^1} = \frac{1}{2}.</math>
- Template:Math is approximately 2.176, which lies between 2 and 3, just as 150 lies between Template:Math and Template:Math.
- For any base Template:Mvar, Template:Math and Template:Math, since Template:Math and Template:Math, respectively.
Logarithmic identitiesEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}
Several important formulas, sometimes called logarithmic identities or logarithmic laws, relate logarithms to one another.<ref>All statements in this section can be found in Template:Harvard citations or Template:Harvard citations, for example.</ref>
Product, quotient, power, and rootEdit
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 Template:Mvar-th power of a number is Template:Mvar times the logarithm of the number itself; the logarithm of a Template:Mvar-th root is the logarithm of the number divided by Template:Mvar. The following table lists these identities with examples. Each of the identities can be derived after substitution of the logarithm definitions <math>x = b^{\, \log_b x}</math> or <math>y = b^{\, \log_b y}</math> in the left hand sides. In the following formulas, Template:Tmath and Template:Tmath are positive real numbers and Template:Tmath is an integer greater than 1.
| Identity | Formula | Example |
|---|---|---|
| Product | <math display="inline">\log_b(x y) = \log_b x + \log_b y</math> | <math display="inline">\log_3 243 = \log_3 (9 \cdot 27) = \log_3 9 + \log_3 27 = 2 + 3 = 5</math> |
| Quotient | <math display="inline">\log_b \!\frac{x}{y} = \log_b x - \log_b y</math> | <math display="inline">\log_2 16 = \log_2 \!\frac{64}{4} = \log_2 64 - \log_2 4 = 6 - 2 = 4</math> |
| Power | <math display="inline">\log_b\left(x^p\right) = p \log_b x</math> | <math display="inline">\log_2 64 = \log_2 \left(2^6\right) = 6 \log_2 2 = 6</math> |
| Root | <math display="inline">\log_b \sqrt[p]{x} = \frac{\log_b x}{p}</math> | <math display="inline">\log_{10} \sqrt{1000} = \frac{1}{2}\log_{10} 1000 = \frac{3}{2} = 1.5</math> |
Change of baseEdit
The logarithm Template:Math can be computed from the logarithms of Template:Mvar and Template:Mvar with respect to an arbitrary base Template:Mvar using the following formula:Template:Refn <math display="block"> \log_b x = \frac{\log_k x}{\log_k b}.</math>
Typical scientific calculators calculate the logarithms to bases 10 and Template:Mvar.<ref>Template:Citation, p. 21</ref> Logarithms with respect to any base Template:Mvar can be determined using either of these two logarithms by the previous formula: <math display="block"> \log_b x = \frac{\log_{10} x}{\log_{10} b} = \frac{\log_{e} x}{\log_{e} b}.</math>
Given a number Template:Mvar and its logarithm Template:Math to an unknown base Template:Mvar, the base is given by:
<math display="block"> b = x^\frac{1}{y},</math>
which can be seen from taking the defining equation <math> x = b^{\,\log_b x} = b^y</math> to the power of <math>\tfrac{1}{y}.</math>
Particular basesEdit
Among all choices for the base, three are particularly common. These are Template:Math, Template:Math (the irrational mathematical constant Template:Nobr and Template:Math (the binary logarithm). In mathematical analysis, the logarithm base Template:Mvar is widespread because of analytical properties explained below. On the other hand, Template:Nobr logarithms (the common logarithm) are easy to use for manual calculations in the decimal number system:<ref> Template:Cite book </ref>
<math display=block>\log_{10}\,(\,10\,x\,)\ =\;\log_{10} 10\ +\;\log_{10} x\ =\ 1\,+\,\log_{10} x\,.</math>
Thus, Template:Math is related to the number of decimal digits of a positive integer Template:Mvar: The number of digits is the smallest integer strictly bigger than Template:Nobr<ref> Template:Cite book </ref> For example, Template:Math 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.<ref> Template:Cite book </ref> Binary logarithms are also used in computer science, where the binary system is ubiquitous; in music theory, where a pitch ratio of two (the octave) 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 (that is, 100 cents per semitone in conventional equal temperament), or equivalently the log base Template:Nobr and in photography rescaled base 2 logarithms are used to measure exposure values, light levels, exposure times, lens apertures, and film speeds in "stops".<ref> Template:Cite book </ref>
The abbreviation Template:Math 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 (base 10), 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 Template:Math still often means the base ten logarithm.<ref> Template:Cite book </ref> In mathematics Template:Math usually refers to the natural logarithm (base Template:Mvar).<ref>Template:Cite book </ref> In computer science and information theory, Template:Math often refers to binary logarithms (base 2).<ref> Template:Cite book </ref> The following table lists common notations for logarithms to these bases. The "ISO notation" column lists designations suggested by the International Organization for Standardization.<ref> Template:Cite report
See also ISO 80000-2 . </ref>
| Base Template:Mvar | Name for logb x | ISO notation | Other notations |
|---|---|---|---|
| 2 | binary logarithm | Template:Math <ref name=gullberg>
Template:Cite book </ref> |
Template:Math, Template:Math, Template:Math,<ref>Template:Citation.</ref> Template:Math |
| Template:Mvar | natural logarithm | Template:Math Template:Refn | Template:Math, Template:Math |
| 10 | common logarithm | Template:Math | Template:Math, Template:Math |
| Template:Mvar | logarithm to base Template:Mvar | Template:Math |
HistoryEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} 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 (Description of the Wonderful Canon of Logarithms).<ref>Template:Citation Template:Pb The sequel ... Constructio was published posthumously: Template:Pb Template:Citation Template:Pb Ian Bruce has made an annotated translation of both books (2012), available from 17centurymaths.com.</ref><ref>Template:Citation</ref> 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.<ref name="folkerts">Template:Citation</ref><ref>Template:Mactutor</ref> Napier coined the term for logarithm in Middle Latin, {{#invoke:Lang|lang}}, literally meaning Template:Gloss, derived from the Greek Template:Transliteration Template:Gloss + Template:Transliteration Template:Gloss.
The common logarithm of a number is the index of that power of ten which equals the number.<ref>William Gardner (1742) Tables of Logarithms</ref> 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".<ref>Template:Citation</ref> 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.<ref>Enrique Gonzales-Velasco (2011) Journey through Mathematics – Creative Episodes in its History, §2.4 Hyperbolic logarithms, p. 117, Springer Template:Isbn</ref> 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 Template:Math was adopted by Gottfried Wilhelm Leibniz in 1675,<ref>Florian Cajori (1913) "History of the exponential and logarithm concepts", American Mathematical Monthly 20: 5, 35, 75, 107, 148, 173, 205</ref> and the next year he connected it to the integral <math display="inline">\int \frac{dy}{y} .</math>
Before Euler developed his modern conception of complex natural logarithms, Roger Cotes had a nearly equivalent result when he showed in 1714 that<ref>Template:Citation</ref> <math display="block">\log(\cos \theta + i\sin \theta) = i\theta.</math>
Logarithm tables, slide rules, and historical applicationsTemplate:AnchorEdit
By simplifying difficult calculations before calculators and computers became available, logarithms contributed to the advance of science, especially astronomy. They were critical to advances in surveying, celestial navigation, and other domains. Pierre-Simon Laplace called logarithms
- "...[a]n admirable artifice which, by reducing to a few days the labour of many months, doubles the life of the astronomer, and spares him the errors and disgust inseparable from long calculations."<ref>Template:Citation, p. 44</ref>
As the function Template:Math is the inverse function of Template:Math, it has been called an antilogarithm.<ref>Template:Citation, section 4.7., p. 89</ref> Nowadays, this function is more commonly called an exponential function.
Log tablesEdit
A key tool that enabled the practical use of logarithms was the table of logarithms.<ref>Template:Citation, section 2</ref> The first such table was compiled by Henry Briggs in 1617, immediately after Napier's invention but with the innovation of using 10 as the base. Briggs' first table contained the common logarithms of all integers in the range from 1 to 1000, with a precision of 14 digits. Subsequently, tables with increasing scope were written. These tables listed the values of Template:Math for any number Template:Mvar in a certain range, at a certain precision. Base-10 logarithms were universally used for computation, hence the name common logarithm, since numbers that differ by factors of 10 have logarithms that differ by integers. The common logarithm of Template:Mvar can be separated into an integer part and a fractional part, known as the characteristic and mantissa. Tables of logarithms need only include the mantissa, as the characteristic can be easily determined by counting digits from the decimal point.<ref>Template:Citation, p. 264</ref> The characteristic of Template:Math is one plus the characteristic of Template:Mvar, and their mantissas are the same. Thus using a three-digit log table, the logarithm of 3542 is approximated by
<math display="block">\begin{align} \log_{10}3542 &= \log_{10}(1000 \cdot 3.542) \\ &= 3 + \log_{10}3.542 \\ &\approx 3 + \log_{10}3.54 \end{align}</math>
Greater accuracy can be obtained by interpolation:
<math display="block"> \log_{10}3542 \approx{} 3 + \log_{10}3.54 + 0.2 (\log_{10}3.55-\log_{10}3.54) </math>
The value of Template:Math can be determined by reverse look up in the same table, since the logarithm is a monotonic function.
ComputationsEdit
The product and quotient of two positive numbers Template:Mvar and Template:Mvar were routinely calculated as the sum and difference of their logarithms. The product Template:Math or quotient Template:Math came from looking up the antilogarithm of the sum or difference, via the same table:
<math display="block"> cd = 10^{\, \log_{10} c} \, 10^{\,\log_{10} d} = 10^{\,\log_{10} c \, + \, \log_{10} d}</math> and <math display="block">\frac c d = c d^{-1} = 10^{\, \log_{10}c \, - \, \log_{10}d}.</math>
For manual calculations that demand any appreciable precision, performing the lookups of the two logarithms, calculating their sum or difference, and looking up the antilogarithm is much faster than performing the multiplication by earlier methods such as prosthaphaeresis, which relies on trigonometric identities.
Calculations of powers and roots are reduced to multiplications or divisions and lookups by <math display="block">c^d = \left(10^{\, \log_{10} c}\right)^d = 10^{\, d \log_{10} c}</math>
and <math display="block">\sqrt[d]{c} = c^\frac{1}{d} = 10^{\frac{1}{d} \log_{10} c}.</math>
Trigonometric calculations were facilitated by tables that contained the common logarithms of trigonometric functions.
Slide rulesEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Another critical application was the slide rule, a pair of logarithmically divided scales used for calculation. The non-sliding logarithmic scale, Gunter's rule, was invented shortly after Napier's invention. William Oughtred enhanced it to create the slide rule—a pair of logarithmic scales movable with respect to each other. Numbers are placed on sliding scales at distances proportional to the differences between their logarithms. Sliding the upper scale appropriately amounts to mechanically adding logarithms, as illustrated here:
For example, adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale yields a product of 6, which is read off at the lower part. The slide rule was an essential calculating tool for engineers and scientists until the 1970s, because it allows, at the expense of precision, much faster computation than techniques based on tables.<ref name="ReferenceA">Template:Citation</ref>
Analytic propertiesEdit
A deeper study of logarithms requires the concept of a function. A function is a rule that, given one number, produces another number.<ref>Template:Citation, or see the references in function</ref> An example is the function producing the Template:Mvar-th power of Template:Mvar from any real number Template:Mvar, where the base Template:Mvar is a fixed number. This function is written as Template:Math. When Template:Mvar is positive and unequal to 1, we show below that Template:Mvar is invertible when considered as a function from the reals to the positive reals.
ExistenceEdit
Let Template:Mvar be a positive real number not equal to 1 and let Template:Math.
It is a standard result in real analysis that any continuous strictly monotonic function is bijective between its domain and range. This fact follows from the intermediate value theorem.<ref name="LangIII.3">Template:Citation, section III.3</ref> Now, Template:Mvar is strictly increasing (for Template:Math), or strictly decreasing (for Template:Math),<ref name="LangIV.2">Template:Harvard citations</ref> is continuous, has domain <math>\R</math>, and has range <math>\R_{> 0}</math>. Therefore, Template:Mvar is a bijection from <math>\R</math> to <math>\R_{>0}</math>. In other words, for each positive real number Template:Mvar, there is exactly one real number Template:Mvar such that <math>b^x = y</math>.
We let <math>\log_b\colon\R_{>0}\to\R</math> denote the inverse of Template:Mvar. That is, Template:Math is the unique real number Template:Mvar such that <math>b^x = y</math>. This function is called the base-Template:Mvar logarithm function or logarithmic function (or just logarithm).
Characterization by the product formulaEdit
The function Template:Math can also be essentially characterized by the product formula <math display="block">\log_b(xy) = \log_b x + \log_b y.</math> More precisely, the logarithm to any base Template:Math is the only increasing function f from the positive reals to the reals satisfying Template:Math and<ref>Template:Citation item (4.3.1)</ref><math display="block">f(xy)=f(x)+f(y).</math>
Graph of the logarithm functionEdit
As discussed above, the function Template:Math is the inverse to the exponential function <math>x\mapsto b^x</math>. Therefore, their graphs correspond to each other upon exchanging the Template:Mvar- and the Template:Mvar-coordinates (or upon reflection at the diagonal line Template:Math), as shown at the right: a point Template:Math on the graph of Template:Mvar yields a point Template:Math on the graph of the logarithm and vice versa. As a consequence, Template:Math diverges to infinity (gets bigger than any given number) if Template:Mvar grows to infinity, provided that Template:Mvar is greater than one. In that case, Template:Math is an increasing function. For Template:Math, Template:Math tends to minus infinity instead. When Template:Mvar approaches zero, Template:Math goes to minus infinity for Template:Math (plus infinity for Template:Math, respectively).
Derivative and antiderivativeEdit
Analytic properties of functions pass to their inverses.<ref name=LangIII.3 /> Thus, as Template:Math is a continuous and differentiable function, so is Template:Math. Roughly, a continuous function is differentiable if its graph has no sharp "corners". Moreover, as the derivative of Template:Math evaluates to Template:Math by the properties of the exponential function, the chain rule implies that the derivative of Template:Math is given by<ref name="LangIV.2"/><ref>Template:Citation</ref> <math display="block">\frac{d}{dx} \log_b x = \frac{1}{x\ln b}. </math> That is, the slope of the tangent touching the graph of the Template:Math logarithm at the point Template:Math equals Template:Math.
The derivative of Template:Math is Template:Math; this implies that Template:Math is the unique antiderivative of Template:Math that has the value 0 for Template:Math. It is this very simple formula that motivated to qualify as "natural" the natural logarithm; this is also one of the main reasons of the importance of the [[E (mathematical constant)|constant Template:Mvar]].
The derivative with a generalized functional argument Template:Math is <math display="block">\frac{d}{dx} \ln f(x) = \frac{f'(x)}{f(x)}.</math> The quotient at the right hand side is called the logarithmic derivative of Template:Mvar. Computing Template:Math by means of the derivative of Template:Math is known as logarithmic differentiation.<ref>Template:Citation, p. 386</ref> The antiderivative of the natural logarithm Template:Math is:<ref>Template:Citation</ref> <math display="block">\int \ln(x) \,dx = x \ln(x) - x + C.</math> Related formulas, such as antiderivatives of logarithms to other bases can be derived from this equation using the change of bases.<ref>Template:Harvard citations</ref>
Integral representation of the natural logarithmEdit
The natural logarithm of Template:Mvar can be defined as the definite integral:
<math display="block">\ln t = \int_1^t \frac{1}{x} \, dx.</math> This definition has the advantage that it does not rely on the exponential function or any trigonometric functions; the definition is in terms of an integral of a simple reciprocal. As an integral, Template:Math equals the area between the Template:Mvar-axis and the graph of the function Template:Math, ranging from Template:Math to Template:Math. This is a consequence of the fundamental theorem of calculus and the fact that the derivative of Template:Math is Template:Math. Product and power logarithm formulas can be derived from this definition.<ref>Template:Citation, section III.6</ref> For example, the product formula Template:Math is deduced as:
<math display="block">\begin{align} \ln(tu) &= \int_1^{tu} \frac{1}{x} \, dx \\ & \stackrel {(1)} = \int_1^{t} \frac{1}{x} \, dx + \int_t^{tu} \frac{1}{x} \, dx \\ & \stackrel {(2)} = \ln(t) + \int_1^u \frac{1}{w} \, dw \\ &= \ln(t) + \ln(u). \end{align}</math>
The equality (1) splits the integral into two parts, while the equality (2) is a change of variable (Template:Math). In the illustration below, the splitting corresponds to dividing the area into the yellow and blue parts. Rescaling the left hand blue area vertically by the factor Template:Mvar and shrinking it by the same factor horizontally does not change its size. Moving it appropriately, the area fits the graph of the function Template:Math again. Therefore, the left hand blue area, which is the integral of Template:Math from Template:Mvar to Template:Mvar is the same as the integral from 1 to Template:Mvar. This justifies the equality (2) with a more geometric proof.
The power formula Template:Math may be derived in a similar way:
<math display="block">\begin{align} \ln(t^r) &= \int_1^{t^r} \frac{1}{x}dx \\ &= \int_1^t \frac{1}{w^r} \left(rw^{r - 1} \, dw\right) \\ &= r \int_1^t \frac{1}{w} \, dw \\ &= r \ln(t). \end{align}</math> The second equality uses a change of variables (integration by substitution), Template:Math.
The sum over the reciprocals of natural numbers, <math display="block">1 + \frac 1 2 + \frac 1 3 + \cdots + \frac 1 n = \sum_{k=1}^n \frac{1}{k},</math> is called the harmonic series. It is closely tied to the natural logarithm: as Template:Mvar tends to infinity, the difference, <math display="block">\sum_{k=1}^n \frac{1}{k} - \ln(n),</math> converges (i.e. gets arbitrarily close) to a number known as the Euler–Mascheroni constant Template:Math. This relation aids in analyzing the performance of algorithms such as quicksort.<ref>Template:Citation, sections 11.5 and 13.8</ref>
Transcendence of the logarithmEdit
Real numbers that are not algebraic are called transcendental;<ref>Template:Citation</ref> for example, [[Pi|Template:Pi]] and e are such numbers, but <math>\sqrt{2-\sqrt 3}</math> is not. Almost all real numbers are transcendental. The logarithm is an example of a transcendental function. The Gelfond–Schneider theorem asserts that logarithms usually take transcendental, i.e. "difficult" values.<ref>Template:Citation, p. 10</ref>
CalculationEdit
Logarithms are easy to compute in some cases, such as Template:Math. In general, logarithms can be calculated using power series or the arithmetic–geometric mean, or be retrieved from a precalculated logarithm table that provides a fixed precision.<ref>Template:Citation, sections 4.2.2 (p. 72) and 5.5.2 (p. 95)</ref><ref>Template:Citation, section 6.3, pp. 105–11</ref> Newton's method, an iterative method to solve equations approximately, can also be used to calculate the logarithm, because its inverse function, the exponential function, can be computed efficiently.<ref>Template:Citation, section 1 for an overview</ref> Using look-up tables, CORDIC-like methods can be used to compute logarithms by using only the operations of addition and bit shifts.<ref>Template:Citation</ref><ref>Template:Citation</ref> Moreover, the binary logarithm algorithm calculates Template:Math recursively, based on repeated squarings of Template:Mvar, taking advantage of the relation <math display="block">\log_2\left(x^2\right) = 2 \log_2 |x|.</math>
Power seriesEdit
Taylor seriesEdit
For any real number Template:Mvar that satisfies Template:Math, the following formula holds:Template:Refn<ref name=AbramowitzStegunp.68>Template:Harvard citations</ref>
<math display="block"> \begin{align}\ln (z) &= \frac{(z-1)^1}{1} - \frac{(z-1)^2}{2} + \frac{(z-1)^3}{3} - \frac{(z-1)^4}{4} + \cdots \\ &= \sum_{k=1}^\infty (-1)^{k+1}\frac{(z-1)^k}{k}. \end{align} </math>
Equating the function Template:Math to this infinite sum (series) is shorthand for saying that the function can be approximated to a more and more accurate value by the following expressions (known as partial sums):
<math display=block> (z-1),\ \ (z-1) - \frac{(z-1)^2}{2},\ \ (z-1) - \frac{(z-1)^2}{2} + \frac{(z-1)^3}{3},\ \ldots </math>
For example, with Template:Math the third approximation yields Template:Math, which is about Template:Math greater than Template:Math, and the ninth approximation yields Template:Math, which is only about Template:Math greater. The Template:Mvarth partial sum can approximate Template:Math with arbitrary precision, provided the number of summands Template:Mvar is large enough.
In elementary calculus, the series is said to converge to the function Template:Math, and the function is the limit of the series. It is the Taylor series of the natural logarithm at Template:Math. The Taylor series of Template:Math provides a particularly useful approximation to Template:Math when Template:Mvar is small, Template:Math, since then <math display="block"> \ln (1+z) = z - \frac{z^2}{2} +\frac{z^3}{3} -\cdots \approx z. </math>
For example, with Template:Math the first-order approximation gives Template:Math, which is less than Template:Math off the correct value Template:Math.
Inverse hyperbolic tangentEdit
Another series is based on the inverse hyperbolic tangent function: <math display="block"> \ln (z) = 2\cdot\operatorname{artanh}\,\frac{z-1}{z+1} = 2 \left ( \frac{z-1}{z+1} + \frac{1}{3}{\left(\frac{z-1}{z+1}\right)}^3 + \frac{1}{5}{\left(\frac{z-1}{z+1}\right)}^5 + \cdots \right ), </math> for any real number Template:Math.Template:Refn<ref name=AbramowitzStegunp.68 /> Using sigma notation, this is also written as <math display="block">\ln (z) = 2\sum_{k=0}^\infty\frac{1}{2k+1}\left(\frac{z-1}{z+1}\right)^{2k+1}.</math> This series can be derived from the above Taylor series. It converges quicker than the Taylor series, especially if Template:Mvar is close to 1. For example, for Template:Math, the first three terms of the second series approximate Template:Math with an error of about Template:Val. The quick convergence for Template:Mvar close to 1 can be taken advantage of in the following way: given a low-accuracy approximation Template:Math and putting <math display="block">A = \frac z{\exp(y)},</math> the logarithm of Template:Mvar is: <math display="block">\ln (z)=y+\ln (A).</math> The better the initial approximation Template:Mvar is, the closer Template:Mvar is to 1, so its logarithm can be calculated efficiently. Template:Mvar can be calculated using the exponential series, which converges quickly provided Template:Mvar is not too large. Calculating the logarithm of larger Template:Mvar can be reduced to smaller values of Template:Mvar by writing Template:Math, so that Template:Math.
A closely related method can be used to compute the logarithm of integers. Putting <math>\textstyle z=\frac{n+1}{n}</math> in the above series, it follows that: <math display="block">\ln (n+1) = \ln(n) + 2\sum_{k=0}^\infty\frac{1}{2k+1}\left(\frac{1}{2 n+1}\right)^{2k+1}.</math> If the logarithm of a large integer Template:Mvar is known, then this series yields a fast converging series for Template:Math, with a rate of convergence of <math display="inline">\left(\frac{1}{2 n+1}\right)^{2}</math>.
Arithmetic–geometric mean approximationEdit
The arithmetic–geometric mean yields high-precision approximations of the natural logarithm. Sasaki and Kanada showed in 1982 that it was particularly fast for precisions between 400 and 1000 decimal places, while Taylor series methods were typically faster when less precision was needed. In their work Template:Math is approximated to a precision of Template:Math (or Template:Mvar precise bits) by the following formula (due to Carl Friedrich Gauss):<ref>Template:Citation</ref><ref>Template:Citation</ref>
<math display="block">\ln (x) \approx \frac{\pi}{2\, \mathrm{M}\!\left(1, 2^{2 - m}/x \right)} - m \ln(2).</math>
Here Template:Math denotes the arithmetic–geometric mean of Template:Mvar and Template:Mvar. It is obtained by repeatedly calculating the average Template:Math (arithmetic mean) and <math display="inline">\sqrt{xy}</math> (geometric mean) of Template:Mvar and Template:Mvar then let those two numbers become the next Template:Mvar and Template:Mvar. The two numbers quickly converge to a common limit which is the value of Template:Math. Template:Mvar is chosen such that
<math display="block">x \,2^m > 2^{p/2}.\, </math>
to ensure the required precision. A larger Template:Mvar makes the Template:Math calculation take more steps (the initial Template:Mvar and Template:Mvar are farther apart so it takes more steps to converge) but gives more precision. The constants Template:Math and Template:Math can be calculated with quickly converging series.
Feynman's algorithmEdit
While at Los Alamos National Laboratory working on the Manhattan Project, Richard Feynman developed a bit-processing algorithm to compute the logarithm that is similar to long division and was later used in the Connection Machine. The algorithm relies on the fact that every real number Template:Mvar where Template:Math can be represented as a product of distinct factors of the form Template:Math. The algorithm sequentially builds that product Template:Mvar, starting with Template:Math and Template:Math: if Template:Math, then it changes Template:Mvar to Template:Math. It then increases <math>k</math> by one regardless. The algorithm stops when Template:Mvar is large enough to give the desired accuracy. Because Template:Math is the sum of the terms of the form Template:Math corresponding to those Template:Mvar for which the factor Template:Math was included in the product Template:Mvar, Template:Math may be computed by simple addition, using a table of Template:Math for all Template:Mvar. Any base may be used for the logarithm table.<ref>Template:Citation</ref>
ApplicationsEdit
Logarithms have many applications inside and outside mathematics. Some of these occurrences are related to the notion of scale invariance. For example, each chamber of the shell of a nautilus is an approximate copy of the next one, scaled by a constant factor. This gives rise to a logarithmic spiral.<ref>Template:Harvard citations</ref> Benford's law on the distribution of leading digits can also be explained by scale invariance.<ref>Template:Citation, chapter 6, section 64</ref> Logarithms are also linked to self-similarity. For example, logarithms appear in the analysis of algorithms that solve a problem by dividing it into two similar smaller problems and patching their solutions.<ref>Template:Citation, p. 21, section 1.3.2</ref> The dimensions of self-similar geometric shapes, that is, shapes whose parts resemble the overall picture are also based on logarithms. Logarithmic scales are useful for quantifying the relative change of a value as opposed to its absolute difference. Moreover, because the logarithmic function Template:Math grows very slowly for large Template:Mvar, logarithmic scales are used to compress large-scale scientific data. Logarithms also occur in numerous scientific formulas, such as the Tsiolkovsky rocket equation, the Fenske equation, or the Nernst equation.
Logarithmic scaleEdit
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Scientific quantities are often expressed as logarithms of other quantities, using a logarithmic scale. For example, the decibel is a unit of measurement associated with logarithmic-scale quantities. It is based on the common logarithm of ratios—10 times the common logarithm of a power ratio or 20 times the common logarithm of a voltage ratio. It is used to quantify the attenuation or amplification of electrical signals,<ref>Template:Cite book</ref> to describe power levels of sounds in acoustics,<ref>Template:Citation, section 23.0.2</ref> and the absorbance of light in the fields of spectrometry and optics. The signal-to-noise ratio describing the amount of unwanted noise in relation to a (meaningful) signal is also measured in decibels.<ref>Template:Citation</ref> In a similar vein, the peak signal-to-noise ratio is commonly used to assess the quality of sound and image compression methods using the logarithm.<ref>Template:Citation</ref>
The strength of an earthquake is measured by taking the common logarithm of the energy emitted at the quake. This is used in the moment magnitude scale or the Richter magnitude scale. For example, a 5.0 earthquake releases 32 times Template:Math and a 6.0 releases 1000 times Template:Math the energy of a 4.0.<ref>Template:Citation, section 4.4.</ref> Apparent magnitude measures the brightness of stars logarithmically.<ref>Template:Citation, section 8.3, p. 231</ref> In chemistry the negative of the decimal logarithm, the decimal Template:Vanchor, is indicated by the letter p.<ref name="Jens">Template:Cite journal</ref> For instance, pH is the decimal cologarithm of the activity of hydronium ions (the form hydrogen ions Template:H+ take in water).<ref>Template:Citation</ref> The activity of hydronium ions in neutral water is 10−7 mol·L−1, hence a pH of 7. Vinegar typically has a pH of about 3. The difference of 4 corresponds to a ratio of 104 of the activity, that is, vinegar's hydronium ion activity is about Template:Math.
Semilog (log–linear) graphs use the logarithmic scale concept for visualization: one axis, typically the vertical one, is scaled logarithmically. For example, the chart at the right compresses the steep increase from 1 million to 1 trillion to the same space (on the vertical axis) as the increase from 1 to 1 million. In such graphs, exponential functions of the form Template:Math appear as straight lines with slope equal to the logarithm of Template:Mvar. Log-log graphs scale both axes logarithmically, which causes functions of the form Template:Math to be depicted as straight lines with slope equal to the exponent Template:Mvar. This is applied in visualizing and analyzing power laws.<ref>Template:Citation, section 34</ref>
PsychologyEdit
Logarithms occur in several laws describing human perception:<ref>Template:Citation, pp. 355–56</ref><ref>Template:Citation, p. 48</ref> Hick's law proposes a logarithmic relation between the time individuals take to choose an alternative and the number of choices they have.<ref>Template:Citation, p. 61</ref> Fitts's law predicts that the time required to rapidly move to a target area is a logarithmic function of the ratio between the distance to a target and the size of the target.<ref>Template:Citation, reprinted in Template:Citation</ref> In psychophysics, the Weber–Fechner law proposes a logarithmic relationship between stimulus and sensation such as the actual vs. the perceived weight of an item a person is carrying.<ref>Template:Citation</ref> (This "law", however, is less realistic than more recent models, such as Stevens's power law.<ref>Template:Citation, lemmas Psychophysics and Perception: Overview</ref>)
Psychological studies found that individuals with little mathematics education tend to estimate quantities logarithmically, that is, they position a number on an unmarked line according to its logarithm, so that 10 is positioned as close to 100 as 100 is to 1000. Increasing education shifts this to a linear estimate (positioning 1000 10 times as far away) in some circumstances, while logarithms are used when the numbers to be plotted are difficult to plot linearly.<ref> Template:Citation </ref><ref>Template:Citation</ref>
Probability theory and statisticsEdit
Logarithms arise in probability theory: the law of large numbers dictates that, for a fair coin, as the number of coin-tosses increases to infinity, the observed proportion of heads approaches one-half. The fluctuations of this proportion about one-half are described by the law of the iterated logarithm.<ref>Template:Citation, section 12.9</ref>
Logarithms also occur in log-normal distributions. When the logarithm of a random variable has a normal distribution, the variable is said to have a log-normal distribution.<ref>Template:Citation</ref> Log-normal distributions are encountered in many fields, wherever a variable is formed as the product of many independent positive random variables, for example in the study of turbulence.<ref> Template:Citation</ref>
Logarithms are used for maximum-likelihood estimation of parametric statistical models. For such a model, the likelihood function depends on at least one parameter that must be estimated. A maximum of the likelihood function occurs at the same parameter-value as a maximum of the logarithm of the likelihood (the "log likelihood"), because the logarithm is an increasing function. The log-likelihood is easier to maximize, especially for the multiplied likelihoods for independent random variables.<ref>Template:Citation, section 11.3</ref>
Benford's law describes the occurrence of digits in many data sets, such as heights of buildings. According to Benford's law, the probability that the first decimal-digit of an item in the data sample is Template:Mvar (from 1 to 9) equals Template:Math, regardless of the unit of measurement.<ref>Template:Citation, section 2.1</ref> Thus, about 30% of the data can be expected to have 1 as first digit, 18% start with 2, etc. Auditors examine deviations from Benford's law to detect fraudulent accounting.<ref>Template:Citation</ref>
The logarithm transformation is a type of data transformation used to bring the empirical distribution closer to the assumed one.
Computational complexityEdit
Analysis of algorithms is a branch of computer science that studies the performance of algorithms (computer programs solving a certain problem).<ref name=Wegener>Template:Citation, pp. 1–2</ref> Logarithms are valuable for describing algorithms that divide a problem into smaller ones, and join the solutions of the subproblems.<ref>Template:Citation, p. 143</ref>
For example, to find a number in a sorted list, the binary search algorithm checks the middle entry and proceeds with the half before or after the middle entry if the number is still not found. This algorithm requires, on average, Template:Math comparisons, where Template:Mvar is the list's length.<ref>Template:Citation, section 6.2.1, pp. 409–26</ref> Similarly, the merge sort algorithm sorts an unsorted list by dividing the list into halves and sorting these first before merging the results. Merge sort algorithms typically require a time approximately proportional to Template:Math.<ref>Template:Harvard citations</ref> The base of the logarithm is not specified here, because the result only changes by a constant factor when another base is used. A constant factor is usually disregarded in the analysis of algorithms under the standard uniform cost model.<ref name=Wegener20>Template:Citation</ref>
A function Template:Math is said to grow logarithmically if Template:Math is (exactly or approximately) proportional to the logarithm of Template:Mvar. (Biological descriptions of organism growth, however, use this term for an exponential function.<ref>Template:Citation, chapter 19, p. 298</ref>) For example, any natural number Template:Mvar can be represented in binary form in no more than Template:Math bits. In other words, the amount of memory needed to store Template:Mvar grows logarithmically with Template:Mvar.
Entropy and chaosEdit
Entropy is broadly a measure of the disorder of some system. In statistical thermodynamics, the entropy Template:Mvar of some physical system is defined as <math display="block"> S = - k \sum_i p_i \ln(p_i).\, </math> The sum is over all possible states Template:Mvar of the system in question, such as the positions of gas particles in a container. Moreover, Template:Math is the probability that the state Template:Mvar is attained and Template:Mvar is the Boltzmann constant. Similarly, entropy in information theory measures the quantity of information. If a message recipient may expect any one of Template:Mvar possible messages with equal likelihood, then the amount of information conveyed by any one such message is quantified as Template:Math bits.<ref>Template:Citation, section III.I</ref>
Lyapunov exponents use logarithms to gauge the degree of chaoticity of a dynamical system. For example, for a particle moving on an oval billiard table, even small changes of the initial conditions result in very different paths of the particle. Such systems are chaotic in a deterministic way, because small measurement errors of the initial state predictably lead to largely different final states.<ref>Template:Citation, section 1.9</ref> At least one Lyapunov exponent of a deterministically chaotic system is positive.
FractalsEdit
Logarithms occur in definitions of the dimension of fractals.<ref>Template:Citation</ref> Fractals are geometric objects that are self-similar in the sense that small parts reproduce, at least roughly, the entire global structure. The Sierpinski triangle (pictured) can be covered by three copies of itself, each having sides half the original length. This makes the Hausdorff dimension of this structure Template:Math. Another logarithm-based notion of dimension is obtained by counting the number of boxes needed to cover the fractal in question.
MusicEdit
Logarithms are related to musical tones and intervals. In equal temperament tunings, the frequency ratio depends only on the interval between two tones, not on the specific frequency, or pitch, of the individual tones. In the 12-tone equal temperament tuning common in modern Western music, each octave (doubling of frequency) is broken into twelve equally spaced intervals called semitones. For example, if the note A has a frequency of 440 Hz then the note B-flat has a frequency of 466 Hz. The interval between A and B-flat is a semitone, as is the one between B-flat and B (frequency 493 Hz). Accordingly, the frequency ratios agree: <math display="block">\frac{466}{440} \approx \frac{493}{466} \approx 1.059 \approx \sqrt[12]2.</math>
Intervals between arbitrary pitches can be measured in octaves by taking the Template:Nowrap logarithm of the frequency ratio, can be measured in equally tempered semitones by taking the Template:Nowrap logarithm (Template:Math times the Template:Nowrap logarithm), or can be measured in cents, hundredths of a semitone, by taking the Template:Nowrap logarithm (Template:Math times the Template:Nowrap logarithm). The latter is used for finer encoding, as it is needed for finer measurements or non-equal temperaments.<ref>Template:Citation, chapter 5</ref>
| Interval (the two tones are played at the same time) |
1/12 tone {{#if:1_step_in_72-et_on_C.mid|{{#ifexist:Media:1_step_in_72-et_on_C.mid|<phonos file="1_step_in_72-et_on_C.mid">play</phonos>|{{errorTemplate:Main other|Audio file "1_step_in_72-et_on_C.mid" not found}}Template:Category handler}}}} |
Semitone {{#if:Minor_second_on_C.mid|{{#ifexist:Media:Minor_second_on_C.mid|<phonos file="Minor_second_on_C.mid">play</phonos>|{{errorTemplate:Main other|Audio file "Minor_second_on_C.mid" not found}}Template:Category handler}}}} |
Just major third {{#if:Just_major_third_on_C.mid|{{#ifexist:Media:Just_major_third_on_C.mid|<phonos file="Just_major_third_on_C.mid">play</phonos>|{{errorTemplate:Main other|Audio file "Just_major_third_on_C.mid" not found}}Template:Category handler}}}} |
Major third {{#if:Major_third_on_C.mid|{{#ifexist:Media:Major_third_on_C.mid|<phonos file="Major_third_on_C.mid">play</phonos>|{{errorTemplate:Main other|Audio file "Major_third_on_C.mid" not found}}Template:Category handler}}}} |
Tritone {{#if:Tritone_on_C.mid|{{#ifexist:Media:Tritone_on_C.mid|<phonos file="Tritone_on_C.mid">play</phonos>|{{errorTemplate:Main other|Audio file "Tritone_on_C.mid" not found}}Template:Category handler}}}} |
Octave {{#if:Perfect_octave_on_C.mid|{{#ifexist:Media:Perfect_octave_on_C.mid|<phonos file="Perfect_octave_on_C.mid">play</phonos>|{{errorTemplate:Main other|Audio file "Perfect_octave_on_C.mid" not found}}Template:Category handler}}}} |
|---|---|---|---|---|---|---|
| Frequency ratio <math>r</math> |
<math>2^{\frac 1 {72}} \approx 1.0097</math> | <math>2^{\frac 1 {12}} \approx 1.0595</math> | <math>\tfrac 5 4 = 1.25</math> | <math>\begin{align} 2^{\frac 4 {12}} & = \sqrt[3] 2 \\ & \approx 1.2599 \end{align} </math> | <math>\begin{align} 2^{\frac 6 {12}} & = \sqrt 2 \\ & \approx 1.4142 \end{align} </math> | <math> 2^{\frac {12} {12}} = 2 </math> |
| Number of semitones <math>12 \log_2 r</math> |
<math>\tfrac 1 6</math> | <math>1</math> | <math>\approx 3.8631</math> | <math>4</math> | <math>6</math> | <math>12</math> |
| Number of cents <math>1200 \log_2 r</math> |
<math>16 \tfrac 2 3</math> | <math>100</math> | <math>\approx 386.31</math> | <math>400</math> | <math>600</math> | <math>1200</math> |
Number theoryEdit
Natural logarithms are closely linked to counting prime numbers (2, 3, 5, 7, 11, ...), an important topic in number theory. For any integer Template:Mvar, the quantity of prime numbers less than or equal to Template:Mvar is denoted Template:Math. The prime number theorem asserts that Template:Math is approximately given by <math display="block">\frac{x}{\ln(x)},</math> in the sense that the ratio of Template:Math and that fraction approaches 1 when Template:Mvar tends to infinity.<ref>Template:Citation, theorem 4.1</ref> As a consequence, the probability that a randomly chosen number between 1 and Template:Mvar is prime is inversely proportional to the number of decimal digits of Template:Mvar. A far better estimate of Template:Math is given by the offset logarithmic integral function Template:Math, defined by <math display="block"> \mathrm{Li}(x) = \int_2^x \frac1{\ln(t)} \,dt. </math> The Riemann hypothesis, one of the oldest open mathematical conjectures, can be stated in terms of comparing Template:Math and Template:Math.<ref>Template:Harvard citations</ref> The Erdős–Kac theorem describing the number of distinct prime factors also involves the natural logarithm.
The logarithm of n factorial, Template:Math, is given by <math display="block"> \ln (n!) = \ln (1) + \ln (2) + \cdots + \ln (n).</math> This can be used to obtain Stirling's formula, an approximation of Template:Math for large Template:Mvar.<ref>Template:Citation, chapter 4</ref>
GeneralizationsEdit
Complex logarithmEdit
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All the complex numbers Template:Mvar that solve the equation
<math display="block">e^a=z</math>
are called complex logarithms of Template:Mvar, when Template:Mvar is (considered as) a complex number. A complex number is commonly represented as Template:Math, where Template:Mvar and Template:Mvar are real numbers and Template:Mvar is an imaginary unit, the square of which is −1. Such a number can be visualized by a point in the complex plane, as shown at the right. The polar form encodes a non-zero complex number Template:Mvar by its absolute value, that is, the (positive, real) distance Template:Mvar to the origin, and an angle between the real (Template:Mvar) axis Template:Math and the line passing through both the origin and Template:Mvar. This angle is called the argument of Template:Mvar.
The absolute value Template:Mvar of Template:Mvar is given by
<math display="block">\textstyle r=\sqrt{x^2+y^2}.</math>
Using the geometrical interpretation of sine and cosine and their periodicity in Template:Math, any complex number Template:Mvar may be denoted as
<math display="block">\begin{align} z &= x + iy \\
&= r (\cos \varphi + i \sin \varphi ) \\ &= r (\cos (\varphi + 2k\pi) + i \sin (\varphi + 2k\pi)),
\end{align}</math>
for any integer number Template:Mvar. Evidently the argument of Template:Mvar is not uniquely specified: both Template:Mvar and Template:Math are valid arguments of Template:Mvar for all integers Template:Mvar, because adding Template:Math radians or k⋅360°Template:Refn to Template:Mvar corresponds to "winding" around the origin counter-clock-wise by Template:Mvar turns. The resulting complex number is always Template:Mvar, as illustrated at the right for Template:Math. One may select exactly one of the possible arguments of Template:Mvar as the so-called principal argument, denoted Template:Math, with a capital Template:Math, by requiring Template:Mvar to belong to one, conveniently selected turn, e.g. Template:Math<ref>Template:Citation, Definition 1.6.3</ref> or Template:Math.<ref>Template:Citation, section 5.9</ref> These regions, where the argument of Template:Mvar is uniquely determined are called branches of the argument function.
Euler's formula connects the trigonometric functions sine and cosine to the complex exponential: <math display="block">e^{i\varphi} = \cos \varphi + i\sin \varphi .</math>
Using this formula, and again the periodicity, the following identities hold:<ref>Template:Citation, section 1.2</ref>
<math display="block"> \begin{align} z &= r \left (\cos \varphi + i \sin \varphi\right) \\
&= r \left (\cos(\varphi + 2k\pi) + i \sin(\varphi + 2k\pi)\right) \\
&= r e^{i (\varphi + 2k\pi)} \\
&= e^{\ln(r)} e^{i (\varphi + 2k\pi)} \\
&= e^{\ln(r) + i(\varphi + 2k\pi)} = e^{a_k},
\end{align} </math>
where Template:Math is the unique real natural logarithm, Template:Math denote the complex logarithms of Template:Mvar, and Template:Mvar is an arbitrary integer. Therefore, the complex logarithms of Template:Mvar, which are all those complex values Template:Math for which the Template:Math power of Template:Mvar equals Template:Mvar, are the infinitely many values <math display="block">a_k = \ln (r) + i ( \varphi + 2 k \pi ),</math> for arbitrary integers Template:Mvar.
Taking Template:Mvar such that Template:Math is within the defined interval for the principal arguments, then Template:Math is called the principal value of the logarithm, denoted Template:Math, again with a capital Template:Math. The principal argument of any positive real number Template:Mvar is 0; hence Template:Math is a real number and equals the real (natural) logarithm. However, the above formulas for logarithms of products and powers [[Exponentiation#Failure of power and logarithm identities|do Template:Em generalize]] to the principal value of the complex logarithm.<ref>Template:Citation, theorem 6.1.</ref>
The illustration at the right depicts Template:Math, confining the arguments of Template:Mvar to the interval Template:Open-closed. This way the corresponding branch of the complex logarithm has discontinuities all along the negative real Template:Mvar axis, which can be seen in the jump in the hue there. This discontinuity arises from jumping to the other boundary in the same branch, when crossing a boundary, i.e. not changing to the corresponding Template:Mvar-value of the continuously neighboring branch. Such a locus is called a branch cut. Dropping the range restrictions on the argument makes the relations "argument of Template:Mvar", and consequently the "logarithm of Template:Mvar", multi-valued functions.
Inverses of other exponential functionsEdit
Exponentiation occurs in many areas of mathematics and its inverse function is often referred to as the logarithm. For example, the logarithm of a matrix is the (multi-valued) inverse function of the matrix exponential.<ref>Template:Citation, chapter 11.</ref> Another example is the p-adic logarithm, the inverse function of the p-adic exponential. Both are defined via Taylor series analogous to the real case.<ref>Template:Neukirch ANT, section II.5.</ref> In the context of differential geometry, the exponential map maps the tangent space at a point of a manifold to a neighborhood of that point. Its inverse is also called the logarithmic (or log) map.<ref>Template:Citation</ref>
In the context of finite groups exponentiation is given by repeatedly multiplying one group element Template:Mvar with itself. The discrete logarithm is the integer Template:Mvar solving the equation <math display="block">b^n = x,</math> where Template:Mvar is an element of the group. Carrying out the exponentiation can be done efficiently, but the discrete logarithm is believed to be very hard to calculate in some groups. This asymmetry has important applications in public key cryptography, such as for example in the Diffie–Hellman key exchange, a routine that allows secure exchanges of cryptographic keys over unsecured information channels.<ref>Template:Citation</ref> Zech's logarithm is related to the discrete logarithm in the multiplicative group of non-zero elements of a finite field.<ref>Template:Citation</ref>
Template:AnchorFurther logarithm-like inverse functions include the double logarithm Template:Math, the super- or hyper-4-logarithm (a slight variation of which is called iterated logarithm in computer science), the Lambert W function, and the logit. They are the inverse functions of the double exponential function, tetration, of Template:Math,<ref>Template:Citation</ref> and of the logistic function, respectively.<ref>Template:Citation, p. 357</ref>
Related conceptsEdit
From the perspective of group theory, the identity Template:Math expresses a group isomorphism between positive reals under multiplication and reals under addition. Logarithmic functions are the only continuous isomorphisms between these groups.<ref>Template:Citation, section V.4.1</ref> By means of that isomorphism, the Haar measure (Lebesgue measure) Template:Math on the reals corresponds to the Haar measure Template:Math on the positive reals.<ref>Template:Citation, section 1.4</ref> The non-negative reals not only have a multiplication, but also have addition, and form a semiring, called the probability semiring; this is in fact a semifield. The logarithm then takes multiplication to addition (log multiplication), and takes addition to log addition (LogSumExp), giving an isomorphism of semirings between the probability semiring and the log semiring.
Logarithmic one-forms Template:Math appear in complex analysis and algebraic geometry as differential forms with logarithmic poles.<ref>Template:Citation, section 2</ref>
The polylogarithm is the function defined by <math display="block"> \operatorname{Li}_s(z) = \sum_{k=1}^\infty {z^k \over k^s}. </math> It is related to the natural logarithm by Template:Math. Moreover, Template:Math equals the Riemann zeta function Template:Math.<ref>Template:Dlmf</ref>
See alsoEdit
NotesEdit
ReferencesEdit
External linksEdit
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|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:Logarithm%7CLogarithm.html}} |title = Logarithm |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}