Mathematical function, inverse of an exponential function
Plots of logarithm functions, with three commonly used bases. The special pointslogbb = 1 are indicated by dotted lines, and all curves intersect inlogb 1 = 0.
Inmathematics, thelogarithm of a number is theexponent by which another fixed value, thebase, must be raised to produce that number. For example, the logarithm of1000 to base10 is3, because1000 is10 to the3rd power:1000 = 103 = 10 × 10 × 10. More generally, ifx =by, theny is the logarithm ofx to baseb, writtenlogbx, solog10 1000 = 3. As a single-variable function, the logarithm to baseb is theinverse ofexponentiation with baseb.
Logarithms were introduced byJohn Napier in 1614 as a means of simplifying calculations.[1] They were rapidly adopted bynavigators, scientists, engineers,surveyors, and others to perform high-accuracy computations more easily. Usinglogarithm tables, tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition. This is possible because the logarithm of aproduct is thesum of the logarithms of the factors:
provided thatb,x andy are all positive andb ≠ 1. Theslide rule, also based on logarithms, allows quick calculations without tables, but at lower precision. The present-day notion of logarithms comes fromLeonhard Euler, who connected them to theexponential function in the 18th century, and who also introduced the lettere as the base of natural logarithms.[2]
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, thecomplex logarithm is the multi-valuedinverse of the complex exponential function. Similarly, thediscrete logarithm is the multi-valued inverse of the exponential function in finite groups; it has uses inpublic-key cryptography.
Thegraph of the logarithm base 2 crosses thex-axis atx = 1 and passes through the points(2, 1),(4, 2), and(8, 3), depicting, e.g.,log2(8) = 3 and23 = 8. The graph gets arbitrarily close to they-axis, butdoes not meet it.
Addition,multiplication, andexponentiation are three of the most fundamental arithmetic operations. The inverse of addition issubtraction, and the inverse of multiplication isdivision. Similarly, a logarithm is the inverse operation ofexponentiation. Exponentiation is when a numberb, thebase, is raised to a certain powery, theexponent, to give a valuex; this is denoted
For example, raising2 to the power of3 gives8:
The logarithm of baseb is the inverse operation, that provides the outputy from the inputx. That is, is equivalent to ifb is a positivereal number. (Ifb 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
by whichtables of logarithms allow multiplication and division to be reduced to addition and subtraction, a great aid to calculations before the invention of computers.
Given a positivereal numberb such thatb ≠ 1, thelogarithm of a positive real numberx with respect to base b[nb 1] is the exponent by whichb must be raised to yieldx. In other words, the logarithm ofx to base b is the unique real number y such that.[3]
The logarithm is denoted "logbx" (pronounced as "the logarithm ofx to base b", "thebase-b logarithm ofx", or most commonly "the log, base b, ofx").
An equivalent and more succinct definition is that the functionlogb is theinverse function to the function.
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 thep-th power of a number isp times the logarithm of the number itself; the logarithm of ap-th root is the logarithm of the number divided byp. 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 arepositive real numbers and is an integer greater than 1.
Product, quotient, power, and root identities of logarithms
The logarithmlogbx can be computed from the logarithms ofx andb with respect to an arbitrary base k using the following formula:[nb 2]
Typicalscientific calculators calculate the logarithms to bases 10 ande.[5] Logarithms with respect to any base b can be determined using either of these two logarithms by the previous formula:
Given a numberx and its logarithmy = logbx to an unknown base b, the base is given by:
which can be seen from taking the defining equation to the power of
Overlaid graphs of the logarithm for bases 1 / 2, 2, ande
Among all choices for the base, three are particularly common. These areb = 10,b =e (theirrational mathematical constante ≈ 2.71828183 ), andb = 2 (thebinary logarithm). Inmathematical analysis, the logarithm basee is widespread because of analytical properties explained below. On the other hand,base 10 logarithms (thecommon logarithm) are easy to use for manual calculations in thedecimal number system:[6]
Thus,log10 (x) is related to the number ofdecimal digits of a positive integerx: The number of digits is the smallestinteger strictly bigger than log10 (x) .[7]For example,log10(5986) is approximately 3.78 . Thenext integer above it is 4, which is the number of digits of 5986. Both the natural logarithm and the binary logarithm are used ininformation theory, corresponding to the use ofnats orbits as the fundamental units of information, respectively.[8]Binary logarithms are also used incomputer science, where thebinary system is ubiquitous; inmusic theory, where a pitch ratio of two (theoctave) is ubiquitous and the number ofcents 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 persemitone inconventional equal temperament), or equivalently the log base21/1200 ; and inphotography rescaled base 2 logarithms are used to measureexposure values,light levels,exposure times, lensapertures, andfilm speeds in "stops".[9]
The abbreviationlog x 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 contextslog x still often means the base ten logarithm.[10] In mathematicslog x usually refers to the natural logarithm (basee).[11]In computer science and information theory,log often refers to binary logarithms (base 2).[12] The following table lists common notations for logarithms to these bases. The "ISO notation" column lists designations suggested by theInternational Organization for Standardization.[13]
The history of logarithms in seventeenth-century Europe saw the discovery of a newfunction that extended the realm of analysis beyond the scope of algebraic methods. The method of logarithms was publicly propounded byJohn Napier in 1614, in a book titledMirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Canon of Logarithms).[19][20] Prior to Napier's invention, there had been other techniques of similar scopes, such as theprosthaphaeresis or the use of tables of progressions, extensively developed byJost Bürgi around 1600.[21][22] Napier coined the term for logarithm in Middle Latin,logarithmus, literally meaning'ratio-number', derived from the Greeklogos'proportion, ratio, word' +arithmos'number'.
Thecommon logarithm of a number is the index of that power of ten which equals the number.[23] Speaking of a number as requiring so many figures is a rough allusion to common logarithm, and was referred to byArchimedes as the "order of a number".[24] 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.[25] Such methods are calledprosthaphaeresis.
By simplifying difficult calculations before calculators and computers became available, logarithms contributed to the advance of science, especiallyastronomy. They were critical to advances insurveying,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.[28]
As the functionf(x) =bx is the inverse function oflogbx, it has been called anantilogarithm.[29] Nowadays, this function is more commonly called anexponential function.
A key tool that enabled the practical use of logarithms was thetable of logarithms.[30] The first such table was compiled byHenry Briggs in 1617, immediately after Napier's invention but with the innovation of using 10 as the base. Briggs' first table contained thecommon 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 oflog10x for any number x 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 ofx can be separated into aninteger part and afractional part, known as the characteristic andmantissa. Tables of logarithms need only include the mantissa, as the characteristic can be easily determined by counting digits from the decimal point.[31] The characteristic of10 ·x is one plus the characteristic ofx, and their mantissas are the same. Thus using a three-digit log table, the logarithm of 3542 is approximated by
The product and quotient of two positive numbersc andd were routinely calculated as the sum and difference of their logarithms. The product cd or quotient c/d came from looking up the antilogarithm of the sum or difference, via the same table:
and
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 asprosthaphaeresis, which relies ontrigonometric identities.
Calculations of powers androots are reduced to multiplications or divisions and lookups by
and
Trigonometric calculations were facilitated by tables that contained the common logarithms oftrigonometric functions.
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:
Schematic depiction of a slide rule. Starting from 2 on the lower scale, add the distance to 3 on the upper scale to reach the product 6. The slide rule works because it is marked such that the distance from 1 tox is proportional to the logarithm ofx.
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.[32]
A deeper study of logarithms requires the concept of afunction. A function is a rule that, given one number, produces another number.[33] An example is the function producing thex-th power ofb from any real number x, where the base b is a fixed number. This function is written asf(x) =bx. Whenb is positive and unequal to 1, we show below thatf is invertible when considered as a function from the reals to the positive reals.
Letb be a positive real number not equal to 1 and letf(x) =bx.
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 theintermediate value theorem.[34] Now,f isstrictly increasing (forb > 1), or strictly decreasing (for0 <b < 1),[35] is continuous, has domain, and has range. Therefore,f is a bijection from to. In other words, for each positive real numbery, there is exactly one real numberx such that.
We let denote the inverse off. That is,logby is the unique real numberx such that. This function is called the base-blogarithm function orlogarithmic function (or justlogarithm).
The graph of the logarithm functionlogb (x) (blue) is obtained byreflecting the graph of the functionbx (red) at the diagonal line (x =y).
As discussed above, the functionlogb is the inverse to the exponential function. Therefore, theirgraphs correspond to each other upon exchanging thex- and they-coordinates (or upon reflection at the diagonal linex =y), as shown at the right: a point(t,u =bt) on the graph off yields a point(u,t = logbu) on the graph of the logarithm and vice versa. As a consequence,logb (x)diverges to infinity (gets bigger than any given number) ifx grows to infinity, provided thatb is greater than one. In that case,logb(x) is anincreasing function. Forb < 1,logb (x) tends to minus infinity instead. Whenx approaches zero,logbx goes to minus infinity forb > 1 (plus infinity forb < 1, respectively).
The graph of thenatural logarithm (green) and its tangent atx = 1.5 (black)
Analytic properties of functions pass to their inverses.[34] Thus, asf(x) =bx is a continuous anddifferentiable function, so islogby. Roughly, a continuous function is differentiable if its graph has no sharp "corners". Moreover, as thederivative off(x) evaluates toln(b)bx by the properties of theexponential function, thechain rule implies that the derivative oflogbx is given by[35][37]
That is, theslope of thetangent touching the graph of thebase-b logarithm at the point(x, logb (x)) equals1/(x ln(b)).
The derivative ofln(x) is1/x; this implies thatln(x) is the uniqueantiderivative of1/x that has the value 0 forx = 1. 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 theconstant e.
The derivative with a generalized functional argumentf(x) is
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,ln(t) equals the area between thex-axis and the graph of the function1/x, ranging fromx = 1 tox =t. This is a consequence of thefundamental theorem of calculus and the fact that the derivative ofln(x) is1/x. Product and power logarithm formulas can be derived from this definition.[41] For example, the product formulaln(tu) = ln(t) + ln(u) is deduced as:
The equality (1) splits the integral into two parts, while the equality (2) is a change of variable (w =x/t). 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 t and shrinking it by the same factor horizontally does not change its size. Moving it appropriately, the area fits the graph of the functionf(x) = 1/x again. Therefore, the left hand blue area, which is the integral off(x) fromt totu is the same as the integral from 1 tou. This justifies the equality (2) with a more geometric proof.
A visual proof of the product formula of the natural logarithm
The power formulaln(tr) =r ln(t) may be derived in a similar way:
The logarithm keys (LOG for base 10 and LN for base e) on aTI-83 Plus graphing calculator
Logarithms are easy to compute in some cases, such aslog10 (1000) = 3. In general, logarithms can be calculated usingpower series or thearithmetic–geometric mean, or be retrieved from a precalculatedlogarithm table that provides a fixed precision.[45][46]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.[47] Using look-up tables,CORDIC-like methods can be used to compute logarithms by using only the operations of addition andbit shifts.[48][49] Moreover, thebinary logarithm algorithm calculateslb(x)recursively, based on repeated squarings ofx, taking advantage of the relation
The Taylor series ofln(z) centered atz = 1. The animation shows the first 10 approximations along with the 99th and 100th. The approximations do not converge beyond a distance of 1 from the center.
For any real numberz that satisfies0 <z ≤ 2, the following formula holds:[nb 4][50]
Equating the functionln(z) 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 aspartial sums):
For example, withz = 1.5 the third approximation yields0.4167, which is about0.011 greater thanln(1.5) = 0.405465, and the ninth approximation yields0.40553, which is only about0.0001 greater. Thenth partial sum can approximateln(z) with arbitrary precision, provided the number of summandsn is large enough.
In elementary calculus, the series is said toconverge to the functionln(z), and the function is thelimit of the series. It is theTaylor series of thenatural logarithm atz = 1. The Taylor series ofln(z) provides a particularly useful approximation toln(1 +z) whenz is small,|z| < 1, since then
For example, withz = 0.1 the first-order approximation givesln(1.1) ≈ 0.1, which is less than5% off the correct value0.0953.
This series can be derived from the above Taylor series. It converges quicker than the Taylor series, especially ifz is close to 1. For example, forz = 1.5, the first three terms of the second series approximateln(1.5) with an error of about3×10−6. The quick convergence forz close to 1 can be taken advantage of in the following way: given a low-accuracy approximationy ≈ ln(z) and putting
the logarithm ofz is:
The better the initial approximationy is, the closerA is to 1, so its logarithm can be calculated efficiently.A can be calculated using theexponential series, which converges quickly providedy is not too large. Calculating the logarithm of largerz can be reduced to smaller values ofz by writingz =a · 10b, so thatln(z) = ln(a) +b · ln(10).
A closely related method can be used to compute the logarithm of integers. Putting in the above series, it follows that:
If the logarithm of a large integer n is known, then this series yields a fast converging series forlog(n+1), with arate of convergence of.
Thearithmetic–geometric mean yields high-precision approximations of thenatural 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 workln(x) is approximated to a precision of2−p (orp precise bits) by the following formula (due toCarl Friedrich Gauss):[51][52]
HereM(x,y) denotes thearithmetic–geometric mean ofx andy. It is obtained by repeatedly calculating the average(x +y)/2 (arithmetic mean) and (geometric mean) ofx andy then let those two numbers become the nextx andy. The two numbers quickly converge to a common limit which is the value ofM(x,y).m is chosen such that
to ensure the required precision. A largerm makes theM(x,y) calculation take more steps (the initialx andy are farther apart so it takes more steps to converge) but gives more precision. The constantsπ andln(2) can be calculated with quickly converging series.
While atLos Alamos National Laboratory working on theManhattan Project,Richard Feynman developed a bit-processing algorithm to compute the logarithm that is similar to long division and was later used in theConnection Machine. The algorithm relies on the fact that every real numberx where1 <x < 2 can be represented as a product of distinct factors of the form1 + 2−k. The algorithm sequentially builds that product P, starting withP = 1 andk = 1: ifP · (1 + 2−k) <x, then it changesP toP · (1 + 2−k). It then increases by one regardless. The algorithm stops whenk is large enough to give the desired accuracy. Becauselog(x) is the sum of the terms of the formlog(1 + 2−k) corresponding to thosek for which the factor1 + 2−k was included in the product P,log(x) may be computed by simple addition, using a table oflog(1 + 2−k) for allk. Any base may be used for the logarithm table.[53]
Logarithms have many applications inside and outside mathematics. Some of these occurrences are related to the notion ofscale invariance. For example, each chamber of the shell of anautilus is an approximate copy of the next one, scaled by a constant factor. This gives rise to alogarithmic spiral.[54]Benford's law on the distribution of leading digits can also be explained by scale invariance.[55] Logarithms are also linked toself-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.[56] 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 functionlog(x) grows very slowly for largex, logarithmic scales are used to compress large-scale scientific data. Logarithms also occur in numerous scientific formulas, such as theTsiolkovsky rocket equation, theFenske equation, or theNernst equation.
Scientific quantities are often expressed as logarithms of other quantities, using alogarithmic scale. For example, thedecibel is aunit of measurement associated withlogarithmic-scalequantities. It is based on the common logarithm ofratios—10 times the common logarithm of apower ratio or 20 times the common logarithm of avoltage ratio. It is used to quantify the attenuation or amplification of electrical signals,[57] to describe power levels of sounds inacoustics,[58] and theabsorbance of light in the fields ofspectrometry andoptics. Thesignal-to-noise ratio describing the amount of unwantednoise in relation to a (meaningful)signal is also measured in decibels.[59] In a similar vein, thepeak signal-to-noise ratio is commonly used to assess the quality of sound andimage compression methods using the logarithm.[60]
The strength of an earthquake is measured by taking the common logarithm of the energy emitted at the quake. This is used in themoment magnitude scale or theRichter magnitude scale. For example, a 5.0 earthquake releases 32 times(101.5) and a 6.0 releases 1000 times(103) the energy of a 4.0.[61]Apparent magnitude measures the brightness of stars logarithmically.[62] Inchemistry the negative of the decimal logarithm, the decimalcologarithm, is indicated by the letter p.[63] For instance,pH is the decimal cologarithm of theactivity ofhydronium ions (the formhydrogenionsH+ take in water).[64] The activity of hydronium ions in neutral water is 10−7mol·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 about10−3 mol·L−1.
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 formf(x) =a ·bx appear as straight lines withslope equal to the logarithm ofb.Log-log graphs scale both axes logarithmically, which causes functions of the formf(x) =a ·xk to be depicted as straight lines with slope equal to the exponent k. This is applied in visualizing and analyzingpower laws.[65]
Logarithms occur in several laws describinghuman perception:[66][67]Hick's law proposes a logarithmic relation between the time individuals take to choose an alternative and the number of choices they have.[68]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.[69] Inpsychophysics, theWeber–Fechner law proposes a logarithmic relationship betweenstimulus andsensation such as the actual vs. the perceived weight of an item a person is carrying.[70] (This "law", however, is less realistic than more recent models, such asStevens's power law.[71])
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.[72][73]
Threeprobability density functions (PDF) of random variables with log-normal distributions. The location parameter μ, which is zero for all three of the PDFs shown, is the mean of the logarithm of the random variable, not the mean of the variable itself.Distribution of first digits (in %, red bars) in thepopulation of the 237 countries of the world. Black dots indicate the distribution predicted by Benford's law.
Logarithms also occur inlog-normal distributions. When the logarithm of arandom variable has anormal distribution, the variable is said to have a log-normal distribution.[75] 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.[76]
Logarithms are used formaximum-likelihood estimation of parametricstatistical models. For such a model, thelikelihood function depends on at least oneparameter 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 forindependent random variables.[77]
Benford's law describes the occurrence of digits in manydata 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 isd (from 1 to 9) equalslog10 (d + 1) − log10 (d),regardless of the unit of measurement.[78] 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.[79]
For example, to find a number in a sorted list, thebinary 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,log2 (N) comparisons, whereN is the list's length.[82] Similarly, themerge 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 timeapproximately proportional toN · log(N).[83] 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 standarduniform cost model.[84]
A function f(x) is said togrow logarithmically iff(x) is (exactly or approximately) proportional to the logarithm ofx. (Biological descriptions of organism growth, however, use this term for an exponential function.[85]) For example, anynatural numberN can be represented inbinary form in no more thanlog2N + 1bits. In other words, the amount ofmemory needed to storeN grows logarithmically withN.
Billiards on an ovalbilliard table. Two particles, starting at the center with an angle differing by one degree, take paths that diverge chaotically because ofreflections at the boundary.
Entropy is broadly a measure of the disorder of some system. Instatistical thermodynamics, the entropy S of some physical system is defined as
The sum is over all possible states i of the system in question, such as the positions of gas particles in a container. Moreover,pi is the probability that the state i is attained andk is theBoltzmann constant. Similarly,entropy in information theory measures the quantity of information. If a message recipient may expect any one ofN possible messages with equal likelihood, then the amount of information conveyed by any one such message is quantified aslog2N bits.[86]
Lyapunov exponents use logarithms to gauge the degree of chaoticity of adynamical 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 arechaotic in adeterministic way, because small measurement errors of the initial state predictably lead to largely different final states.[87] At least one Lyapunov exponent of a deterministically chaotic system is positive.
The Sierpinski triangle (at the right) is constructed by repeatedly replacingequilateral triangles by three smaller ones.
Logarithms occur in definitions of thedimension offractals.[88] Fractals are geometric objects that are self-similar in the sense that small parts reproduce, at least roughly, the entire global structure. TheSierpinski triangle (pictured) can be covered by three copies of itself, each having sides half the original length. This makes theHausdorff dimension of this structureln(3)/ln(2) ≈ 1.58. Another logarithm-based notion of dimension is obtained bycounting the number of boxes needed to cover the fractal in question.
Four different octaves shown on a linear scale, then shown on a logarithmic scale (as the ear hears them)
Logarithms are related to musical tones andintervals. Inequal temperament tunings, the frequency ratio depends only on the interval between two tones, not on the specific frequency, orpitch, of the individual tones. In the12-tone equal temperament tuning common in modern Western music, eachoctave (doubling of frequency) is broken into twelve equally spaced intervals calledsemitones. For example, if thenote A has a frequency of 440 Hz then the noteB-flat has a frequency of 466 Hz. The interval betweenA andB-flat is asemitone, as is the one betweenB-flat andB (frequency 493 Hz). Accordingly, the frequency ratios agree:
Intervals between arbitrary pitches can be measured in octaves by taking thebase-2 logarithm of thefrequency ratio, can be measured in equally tempered semitones by taking thebase-21/12 logarithm (12 times thebase-2 logarithm), or can be measured incents, hundredths of a semitone, by taking thebase-21/1200 logarithm (1200 times thebase-2 logarithm). The latter is used for finer encoding, as it is needed for finer measurements or non-equal temperaments.[89]
Interval (the two tones are played at the same time)
in the sense that the ratio ofπ(x) and that fraction approaches 1 whenx tends to infinity.[90] As a consequence, the probability that a randomly chosen number between 1 andx is prime is inverselyproportional to the number of decimal digits ofx. A far better estimate ofπ(x) is given by theoffset logarithmic integral functionLi(x), defined by
are calledcomplex logarithms ofz, whenz is (considered as) a complex number. A complex number is commonly represented asz = x + iy, wherex andy are real numbers andi is animaginary unit, the square of which is −1. Such a number can be visualized by a point in thecomplex plane, as shown at the right. Thepolar form encodes a non-zero complex number z by itsabsolute value, that is, the (positive, real) distance r to theorigin, and an angle between the real (x) axisRe and the line passing through both the origin andz. This angle is called theargument ofz.
The absolute valuer ofz is given by
Using the geometrical interpretation ofsine andcosine and their periodicity in2π, any complex number z may be denoted as
for any integer number k. Evidently the argument ofz is not uniquely specified: bothφ andφ' =φ + 2kπ are valid arguments ofz for all integers k, because adding2kπradians ork⋅360°[nb 6] toφ corresponds to "winding" around the origin counter-clock-wise bykturns. The resulting complex number is alwaysz, as illustrated at the right fork = 1. One may select exactly one of the possible arguments ofz as the so-calledprincipal argument, denotedArg(z), with a capital A, by requiringφ to belong to one, conveniently selected turn, e.g.−π <φ ≤π[93] or0 ≤φ < 2π.[94] These regions, where the argument ofz is uniquely determined are calledbranches of the argument function.
The principal branch (-π,π) of the complex logarithm,Log(z). The black point atz = 1 corresponds to absolute value zero and brighter colors refer to bigger absolute values. Thehue of the color encodes the argument ofLog(z).
Using this formula, and again the periodicity, the following identities hold:[95]
whereln(r) is the unique real natural logarithm,ak denote the complex logarithms ofz, andk is an arbitrary integer. Therefore, the complex logarithms ofz, which are all those complex valuesak for which theak-th power ofe equalsz, are the infinitely many values
for arbitrary integersk.
Takingk such thatφ + 2kπ is within the defined interval for the principal arguments, thenak is called theprincipal value of the logarithm, denotedLog(z), again with a capital L. The principal argument of any positive real number x is 0; henceLog(x) is a real number and equals the real (natural) logarithm. However, the above formulas for logarithms of products and powersdonot generalize to the principal value of the complex logarithm.[96]
The illustration at the right depictsLog(z), confining the arguments ofz to the interval(−π, π]. This way the corresponding branch of the complex logarithm has discontinuities all along the negative realx 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 correspondingk-value of the continuously neighboring branch. Such a locus is called abranch cut. Dropping the range restrictions on the argument makes the relations "argument ofz", and consequently the "logarithm ofz",multi-valued functions.
In the context offinite groups exponentiation is given by repeatedly multiplying one group element b with itself. Thediscrete logarithm is the integer n solving the equation
wherex 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 inpublic key cryptography, such as for example in theDiffie–Hellman key exchange, a routine that allows secure exchanges ofcryptographic keys over unsecured information channels.[100]Zech's logarithm is related to the discrete logarithm in the multiplicative group of non-zero elements of afinite field.[101]
From the perspective ofgroup theory, the identitylog(cd) = log(c) + log(d) expresses agroup isomorphism between positivereals under multiplication and reals under addition. Logarithmic functions are the only continuous isomorphisms between these groups.[104] By means of that isomorphism, theHaar measure (Lebesgue measure) dx on the reals corresponds to the Haar measure dx/x on the positive reals.[105] The non-negative reals not only have a multiplication, but also have addition, and form asemiring, called theprobability semiring; this is in fact asemifield. The logarithm then takes multiplication to addition (log multiplication), and takes addition to log addition (LogSumExp), giving anisomorphism of semirings between the probability semiring and thelog semiring.
^Proof:Taking the logarithm to basek of the defining identity one getsThe formula follows by solving for
^zSome mathematicians disapprove of this notation. In his 1985 autobiography,Paul Halmos criticized what he considered the "childishln notation", which he said no mathematician had ever used.[16] The notation was invented by the 19th century mathematicianI. Stringham.[17][18]
^The same series holds for the principal value of the complex logarithm for complex numbersz satisfying|z − 1| < 1.
^The same series holds for the principal value of the complex logarithm for complex numbersz with positive real part.
^Wegener, Ingo (2005),Complexity Theory: Exploring the limits of efficient algorithms, Berlin, DE / New York, NY:Springer-Verlag, p. 20,ISBN978-3-540-21045-0
^Goodrich, Michael T.;Tamassia, Roberto (2002),Algorithm Design: Foundations, analysis, and internet examples, John Wiley & Sons, p. 23,One of the interesting and sometimes even surprising aspects of the analysis of data structures and algorithms is the ubiquitous presence of logarithms ... As is the custom in the computing literature, we omit writing the baseb of the logarithm whenb = 2 .
^Enrique Gonzales-Velasco (2011)Journey through Mathematics – Creative Episodes in its History, §2.4 Hyperbolic logarithms, p. 117, SpringerISBN978-0-387-92153-2
^Spiegel, Murray R.; Moyer, R.E. (2006),Schaum's outline of college algebra, Schaum's outline series, New York:McGraw-Hill,ISBN978-0-07-145227-4, p. 264
^Hart; Cheney; Lawson; et al. (1968), "Computer Approximations",Physics Today, SIAM Series in Applied Mathematics,21 (2), New York: John Wiley: 91,Bibcode:1968PhT....21b..91D,doi:10.1063/1.3034795, section 6.3, pp. 105–11
^Meggitt, J. E. (April 1962), "Pseudo Division and Pseudo Multiplication Processes",IBM Journal of Research and Development,6 (2):210–26,doi:10.1147/rd.62.0210,S2CID19387286
^Kahan, W. (20 May 2001),Pseudo-Division Algorithms for Floating-Point Logarithms and Exponentials
^Ahrendt, Timm (1999), "Fast Computations of the Exponential Function",Stacs 99, Lecture notes in computer science, vol. 1564, Berlin, New York: Springer, pp. 302–12,doi:10.1007/3-540-49116-3_28,ISBN978-3-540-65691-3
^Sankaran, C. (2001), "7.5.1 Decibel (dB)",Power Quality, Taylor & Francis,ISBN9780849310409,The decibel is used to express the ratio between two quantities. The quantities may be voltage, current, or power.
^Maling, George C. (2007), "Noise", in Rossing, Thomas D. (ed.),Springer handbook of acoustics, Berlin, New York:Springer-Verlag,ISBN978-0-387-30446-5, section 23.0.2
^Crauder, Bruce; Evans, Benny; Noell, Alan (2008),Functions and Change: A Modeling Approach to College Algebra (4th ed.), Boston: Cengage Learning,ISBN978-0-547-15669-9, section 4.4.
^Rose, Colin; Smith, Murray D. (2002),Mathematical statistics with Mathematica, Springer texts in statistics, Berlin, New York:Springer-Verlag,ISBN978-0-387-95234-5, section 11.3
^Cherkassky, Vladimir; Cherkassky, Vladimir S.; Mulier, Filip (2007),Learning from data: concepts, theory, and methods, Wiley series on adaptive and learning systems for signal processing, communications, and control, New York:John Wiley & Sons,ISBN978-0-471-68182-3, p. 357
