Wikipedia; Logarithms

8/10/2019 Wikipedia; Logarithms

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Logarithm

Thegraph of the logarithm to base 2 crosses the x axis(horizontalaxis) at 1 and passes through the points with coordinates(2, 1),

(4, 2), and (8, 3). For example, log2(8) = 3, because 23 = 8.

The graph gets arbitrarily close to the yaxis, butdoes not meet orintersect it.

In mathematics, the logarithm of a number is theexponentto which another fixed value, thebase, must be

raised to produce that number. For example, the loga-rithm of 1000 to base 10 is 3, because 10 to the power 3is 1000: 1000 = 10 10 10 = 103. More generally, forany tworeal numbersbandxwherebis positive andb1,

y= bx x= logb(y)

The logarithm to base 10 (b= 10) is called thecommonlogarithmand has many applications in science and engi-neering. Thenatural logarithmhas theirrational (tran-scendental) number e ( 2.718) as its base; its use is

widespread inpure mathematics, especially calculus. Thebinary logarithmuses base 2 (b= 2) and is prominent incomputer science.

Logarithms were introduced byJohn Napierin the early17th century as a means to simplify calculations. Theywere rapidly adopted by navigators, scientists, engineers,and others to perform computations more easily, usingslide rulesandlogarithm tables. Tedious multi-digit mul-tiplication steps can be replaced by table look-ups andsimpler addition because of the factimportant in itsown rightthat the logarithm of aproductis thesumofthe logarithms of the factors:

logb(xy) =logb(x) +logb(y),

provided that b, xand y are all positive and b 1. Thepresent-day notion of logarithms comes fromLeonhardEuler, who connected them to theexponential functioninthe 18th century.

Logarithmic scales reduce wide-ranging quantities tosmaller scopes. For example, the decibel is a loga-rithmic unit quantifying signal power ratios and ampli-tude ratios (of whichsound pressureis a common exam-ple). In chemistry,pHis a logarithmic measure for theacidityof anaqueous solution. Logarithms are common-

place in scientificformulae, and in measurements of thecomplexity of algorithmsand of geometric objects calledfractals. They describemusical intervals, appear in for-mulae countingprime numbers, inform some models inpsychophysics, and can aid inforensic accounting.

In the same way as the logarithm reverses exponentiation,thecomplex logarithmis theinverse functionof the ex-ponential function applied to complex numbers. Thediscrete logarithmis another variant; it has applicationsinpublic-key cryptography.

1 Motivation and definition

The idea of logarithms is to reverse the operation ofexponentiation, that is raising a number to a power. Forexample, the third power (orcube) of 2 is 8, because 8 isthe product of three factors of 2:

23 = 2 2 2 = 8.

It follows that the logarithm of 8 with respect to base 2 is3, so log28 = 3.

1.1 Exponentiation

The third power of some number bis the product of threefactors ofb. More generally, raisingbto then-th power,wherenis anatural number, is done by multiplyingnfac-tors ofb. Then-th power ofbis writtenbn, so that

bn =b b b

nfactors.

Exponentiation may be extended to by, whereb is a pos-itive number and the exponent yis anyreal number. For

1
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2 3 PARTICULAR BASES

example,b1 is thereciprocalofb, that is, 1/b. (For fur-ther details, including the formula bm+ n = bm bn, seeexponentiationor [1] for an elementary treatise.)

1.2 Definition

Thelogarithmof a positive real numberxwith respect tobaseb, a positive real number not equal to 1[nb 1], is theexponent by which b must be raised to yield x. In otherwords, the logarithm ofxto basebis the solutionyto theequation[2]

by =x.

The logarithm is denoted logb(x)" (pronounced as thelogarithm ofxto baseb" or the base-blogarithm ofx").In the equation y = logb(x), the value y is the answer to

the question To what power must b be raised, in orderto yield x?". This question can also be addressed (witha richer answer) forcomplex numbers, which is done insectionComplex logarithm, and this answer is muchmore extensively investigated inthe page for the complexlogarithm.

1.3 Examples

For example, log2(16) = 4, since 24 = 2 2 2 2 = 16.Logarithms can also be negative:

log2

1

2

= 1,

since

21 = 1

21 =

1

2.

A third example: log10(150) is approximately 2.176,which lies between 2 and 3, just as 150 lies between 102

= 100 and 103 = 1000. Finally, for any baseb, logb(b) =

1 and logb(1) = 0, sinceb1

=bandb0

= 1, respectively.

2 Logarithmic identities

Main article:List of logarithmic identities

Several important formulas, sometimes called logarithmicidentitiesorlog laws, relate logarithms to one another.[3]

2.1 Product, quotient, power and root

The logarithm of a product is the sum of the logarithms ofthe numbers being multiplied; the logarithm of the ratio

of two numbers is the difference of the logarithms. Thelogarithm of the p-th power of a number is p times thelogarithm of the number itself; the logarithm of a p-throot is the logarithm of the number divided by p. Thefollowing table lists these identities with examples. Eachof the identities can be derived after substitution of the

logarithm definitions x = blog(x), and/or y = blog(y), in theleft hand sides.

2.2 Change of base

The logarithm logb(x) can be computed from the loga-rithms ofxandbwith respect to an arbitrary basekusingthe following formula:

logb(x) = log

k(x)

logk(b) .

Typical scientific calculatorscalculate the logarithms tobases 10 ande.[4] Logarithms with respect to any basebcan be determined using either of these two logarithmsby the previous formula:

logb(x) = log10(x)

log10(b) =

loge(x)

loge(b).

Given a numberxand its logarithm logb(x)toanunknownbaseb, the base is given by:

b= x1

logb(x) .

3 Particular bases

Among all choices for the base, three are particularlycommon. These are b = 10,b = e (theirrationalmathe-matical constant 2.71828), andb= 2. Inmathematicalanalysis, the logarithm to baseeis widespread because ofits particular analytical properties explained below. Onthe other hand, base-10 logarithms are easy to use formanual calculations in thedecimalnumber system:[5]

log10(10x) =log10(10) +log10(x) = 1 +log10(x).

Thus, log10(x) is related to the number of decimal dig-itsof a positive integer x: the number of digits is thesmallest integerstrictly bigger than log10(x).

[6] For ex-ample, log10(1430) is approximately 3.15. The nextinteger is 4, which is the number of digits of 1430.Both the natural logarithm and the logarithm to base

two are used ininformation theory, corresponding to theuse of nats or bits as the fundamental units of infor-mation, respectively.[7] Binary logarithms are also used
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4.2 From Napier to Euler 3

incomputer science, where thebinary systemis ubiq-uitous, inmusic theory, where a pitch ratio of two (theoctave) is ubiquitous and the cent is the binary loga-rithm (scaled by 1200) of the ratio between two adjacentequally-tempered pitches, and in photography to measureexposure values.[8]

The following table lists common notations for logarithmsto these bases and the fields where they are used. Manydisciplines write log(x) instead of logb(x), when the in-tended base can be determined from the context. Thenotation blog(x) also occurs.[9] The ISO notation col-umn lists designations suggested by theInternational Or-ganization for Standardization(ISO 31-11).[10]

4 History

4.1 Predecessors

TheBabylonianssometime in 20001600 BC may haveinvented thequarter square multiplication algorithm tomultiply two numbers using only addition, subtractionand a table of quarter squares.[15][16] However, it couldnot be used for division without an additional table ofreciprocals (or the knowledge of a sufficiently simplealgorithm to generate reciprocals). Large tables of quar-ter squares were used to simplify the accurate multiplica-

tion of large numbers from 1817 onwards until this wassuperseded by the use of computers.

The Indian mathematician Virasena worked with the con-cept of ardhaccheda: the number of times a number ofthe form 2n could be halved. For exact powers of 2, thisis the logarithm to that base, which is a whole number;for other numbers, it is undefined. He described rela-tions such as the product formula and also introducedinteger logarithms in base 3 (trakacheda) and base 4(caturthacheda)[17]

Michael Stifel published Arithmetica integra inNuremberg in 1544, which contains a table[18] of

integers and powers of 2 that has been considered anearly version of a logarithmic table.[19][20]

In the 16th and early 17th centuries an algorithm calledprosthaphaeresiswas used to approximate multiplicationand division. This used the trigonometric identity

cos cos = 1

2[cos(+ ) +cos( )]

or similar to convert the multiplications to additions and

table lookups. However, logarithms are more straightfor-ward and require less work. It can be shown usingEulersFormulathat the two techniques are related.

John Napier (15501617), the inventor of logarithms

4.2 From Napier to Euler

The method of logarithms was publicly propounded byJohn Napier in 1614, in a book titled Mirifici Loga-rithmorum Canonis Descriptio

(Description of the Won-

derful Rule of Logarithms).[21][22] Joost Brgi indepen-dently invented logarithms but published six years afterNapier.[23][24]

Johannes Kepler, who used logarithm tables extensivelyto compile his Ephemerisand therefore dedicated it toNapier,[25] remarked:

the accent in calculation led JustusByrgius [Joost Brgi] on the way to these verylogarithms many years before Napiers system

appeared; but instead of rearing up hischild for the public benefit he deserted it in thebirth.Johannes Kepler[26], Rudolphine Tables(1627)

By repeated subtractions Napier calculated (1 107)L

forL ranging from 1 to 100. The result for L=100 is ap-proximately 0.99999 = 1 105. Napier then calculatedthe products of these numbers with 107(1 105)L forLfrom 1 to50, and did similarly with0.9998 (1 105)20

and 0.9 0.99520. These computations, which occupied20 years, allowed him to give, for any numberNfrom 5to 10 million, the numberLthat solves the equation
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4 4 HISTORY

N= 107(1 107)L.

Napier first called L an artificial number, but later in-troduced the wordlogarithmto mean a number that in-

dicates a ratio: (logos) meaning proportion, and (arithmos) meaning number. In modern nota-tion, the relation tonatural logarithmsis: [27]

L= log(1107)

N

107

107 log1

e

N

107

= 107 loge

N

107

,

where the very close approximation corresponds to theobservation that

(1 107

)107

1

e .

The invention was quickly and widely met with acclaim.The works of Bonaventura Cavalieri (Italy), EdmundWingate(France), Xue Fengzuo (China), andJohannesKepler'sChilias logarithmorum(Germany) helped spreadthe concept further.[28]

The hyperbolay= 1/x(red curve) and the area fromx= 1 to 6(shaded in orange).

In 1649,Alphonse Antonio de Sarasa, a former studentof Grgoire de Saint-Vincent,[29] related logarithms to thequadrature of the hyperbola, by pointing out that the area

f(t) under the hyperbola from x= 1 tox= tsatisfies[30]

f(tu) =f(t) + f(u).

The natural logarithmwas first described by NicholasMercator in his work Logarithmotechnia published in1668,[31] although the mathematics teacher John Speidell

had already in 1619 compiled a table of what were ef-fectivelynatural logarithms, based on Napiers work.[32]

Around 1730, Leonhard Euler defined the exponentialfunction and thenatural logarithmby

ex = limn

(1 + x/n)n,

ln(x) = limn

n(x1/n 1).

Euler also showed that the two functions are inverse toone another.[33][34][35]

4.3 Logarithm tables, slide rules, and his-

torical applications

The 1797Encyclopdia Britannicaexplanation of logarithms

By simplifying difficult calculations, logarithms con-tributed to the advance of science, and especially ofastronomy. They were critical to advances insurveying,

celestial navigation, and other domains. Pierre-SimonLaplacecalled logarithms

"...[a]n admirable artifice which, byreducing to a few days the labourof many months, doubles the life ofthe astronomer, and spares him theerrors and disgust inseparable fromlong calculations.[36]

A key tool that enabled the practical use of logarithmsbefore calculators and computers was the table of log-arithms.[37] The first such table was compiled byHenryBriggs in 1617, immediately after Napiers invention.Subsequently, tables with increasing scope and precisionwere written. These tables listed the values of logb(x) andbx for any number xin a certain range, at a certain pre-cision, for a certain base b (usually b = 10). For exam-ple, Briggs first table contained the common logarithmsof all integers in the range 11000, with a precision of8 digits. As the function f(x) = bx is the inverse func-tion of logb(x), it has been called the antilogarithm.[38]

The product and quotient of two positive numbers canddwere routinely calculated as the sum and difference of

their logarithms. The product cdor quotient c/dcamefrom looking up the antilogarithm of the sum or differ-ence, also via the same table:
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5

cd= blogb(c) blogb(d) =blogb(c)+logb(d)

and

cd

=cd1 =blogb(c)logb(d).

For manual calculations that demand any appreciable pre-cision, performing the lookups of the two logarithms, cal-culating their sum or difference, and looking up the an-tilogarithm is much faster than performing the multipli-cation by earlier methods such as prosthaphaeresis, whichrelies ontrigonometric identities. Calculations of powersandrootsare reduced to multiplications or divisions andlook-ups by

cd

= (blog

b(c)

)d

=bd log

b(c)

and

d

c= c1d =b

1d

logb(c).

Many logarithm tables give logarithms by separately pro-viding the characteristic andmantissaofx, that is to say,the integer part and the fractional part of log10(x).

[39] Thecharacteristic of 10 xis one plus the characteristic of

x, and theirsignificandsare the same. This extends thescope of logarithm tables: given a table listing log10(x)

for all integers xranging from 1 to 1000, the logarithmof 3542 is approximated by

log10(3542) =log10(10354.2) = 1+log10(354.2) 1+log10(354).

Another critical application was theslide rule, a pair oflogarithmically divided scales used for calculation, as il-lustrated here:

1 2 3 4 5 6 7 8 9 1

1 2 3 4 5 6 7 8 9 1

log(2)

log(3)

log(6)

Schematic depiction of a slide rule. Starting from 2 on the lower

scale, add the distanceto 3 onthe upper scale toreach the product

6. Thesliderule works because it is markedsuch that the distance

from 1 tox is proportional to the logarithm ofx.

The non-sliding logarithmic scale, Gunters rule, wasinvented shortly after Napiers invention. WilliamOughtredenhanced it to create the slide rulea pair oflogarithmic scales movable with respect to each other.Numbers are placed on sliding scales at distances propor-tional to the differences between their logarithms. Slid-

ing the upper scale appropriately amounts to mechani-cally adding logarithms. For example, adding the dis-tance 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 readoff at the lower part. The slide rule was an essential cal-culating tool for engineers and scientists until the 1970s,because it allows, at the expense of precision, much fastercomputation than techniques based on tables.[33]

5 Analytic properties

A deeper study of logarithms requires the concept of afunction. A function is a rule that, given one number,produces another number.[40] An example is the functionproducing the x-th power of b from any real number x,where the basebis a fixed number. This function is writ-ten

f(x) =bx.

5.1 Logarithmic function

To justify the definition of logarithms, it is necessary toshow that the equation

bx =y

has a solutionxand that this solution is unique, providedthatyis positive and thatbis positive and unequal to 1. A

proof of that fact requires the intermediate value theoremfrom elementarycalculus.[41] This theorem states that acontinuous function that produces two values m and n alsoproduces any value that lies between mandn. A functioniscontinuousif it does not jump, that is, if its graph canbe drawn without lifting the pen.

This property can be shown to hold for the functionf(x)= bx. Because f takes arbitrarily large and arbitrarilysmall positive values, any number y > 0 lies betweenf(x0)andf(x1) for suitable x0andx1. Hence, the intermedi-ate value theorem ensures that the equation f(x) = y hasa solution. Moreover, there is only one solution to this

equation, because the functionfisstrictly increasing(forb> 1), or strictly decreasing (for 0 < b< 1).[42]

The unique solution xis the logarithm of y to base b,logb(y). The function that assigns to y its logarithm iscalledlogarithm functionor logarithmic function(or justlogarithm).

The function logb(x) is essentially characterized by theabove product formula

logb(xy) =logb(x) +logb(y).

More precisely, the logarithm to any base b >1istheonlyincreasing functionffrom the positive reals to the realssatisfyingf(b) = 1 and [43]
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6 5 ANALYTIC PROPERTIES

f(xy) =f(x) + f(y).

5.2 Inverse function

logb(t) tx

logb(t)

t

bx

logb(x)

The graph of the logarithm function log(x) (blue) is obtained byreflectingthe graph of the function bx (red) at the diagonal line(x=y).

The formula for the logarithm of a power says in partic-ular that for any numberx,

logb(bx) =x logb(b) =x.

In prose, taking the x-th power of b and then the base-b logarithm gives back x. Conversely, given a positivenumbery, the formula

blogb(y) =y

says that first taking the logarithm and then exponentiat-ing gives backy. Thus, the two possible ways of combin-ing (or composing) logarithms and exponentiation giveback the original number. Therefore, the logarithm tobasebis the inverse functionoff(x) = bx.[44]

Inverse functions are closely related to the original func-tions. Theirgraphscorrespond to each other upon ex-changing thex- and they-coordinates (or upon reflectionat the diagonal linex= y), as shown at the right: a point(t,u = bt) on the graph of fyields a point (u,t= logbu)

on the graph of the logarithm and vice versa. As a conse-quence, logb(x)diverges to infinity(gets bigger than anygiven number) if xgrows to infinity, provided that b is

greater than one. In that case, logb(x) is an increasingfunction. Forb< 1, logb(x) tends to minus infinity in-stead. When xapproaches zero, logb(x) goes to minusinfinity forb> 1 (plus infinity forb< 1, respectively).

5.3 Derivative and antiderivative

The graph of thenatural logarithm(green) and its tangent atx= 1.5 (black)

Analytic properties of functions pass to their inverses.[41]

Thus, asf(x) =bx is a continuous anddifferentiable func-tion, so is logb(y). Roughly, a continuous function is dif-ferentiable if its graph has no sharp corners. Moreover,as thederivativeoff(x) evaluates to ln(b)bx by the prop-erties of theexponential function, thechain ruleimpliesthat the derivative of logb(x) is given by[42][45]

d

dxlogb(x) =

1

x ln(b).

That is, the slopeof thetangent touching the graph ofthe base-blogarithm at the point (x, logb(x)) equals 1/(xln(b)). In particular, the derivative of ln(x) is 1/x, whichimplies that theantiderivativeof 1/x is ln(x) + C. Thederivative with a generalised functional argument f(x) is

ddx

ln(f(x)) = f

(x)f(x)

.

The quotient at the right hand side is called thelogarithmic derivativeoff. Computingf'(x) by means ofthe derivative of ln(f(x)) is known aslogarithmic differ-entiation.[46] The antiderivative of thenatural logarithmln(x) is:[47]

ln(x) dx= x ln(x) x + C.

Related formulas, such as antiderivatives of logarithms toother bases can be derived from this equation using thechange of bases.[48]
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5.5 Transcendence of the logarithm 7

5.4 Integral representation of thenatural

logarithm

Thenatural logarithm of t is the shaded area underneath thegraph of the functionf(x) = 1/x(reciprocal ofx).

Thenatural logarithmoftagrees with theintegralof 1/xdxfrom 1 tot:

ln(t) =t1

1x

dx.

In other words, ln(t) equals the area between the xaxisand the graph of the function 1/x, ranging fromx= 1 to

x= t(figure at the right). This is a consequence of thefundamental theorem of calculusand the fact that deriva-tive of ln(x) is 1/x. The right hand side of this equationcan serve as a definition of thenatural logarithm. Productand power logarithm formulas can be derived from thisdefinition.[49] For example, the product formula ln(tu) =ln(t) + ln(u) is deduced as:

ln(tu) =

tu1

1

xdx

(1)=

t1

1

xdx+

tut

1

xdx

(2)=ln(t)+

u1

1

wdw = ln(t)+ln(u).

The equality (1) splits the integral into two parts, whilethe equality (2) is a change of variable (w= x/t). In theillustration below, the splitting corresponds to dividingthe area into the yellow and blue parts. Rescaling theleft hand blue area vertically by the factor tand shrink-ing it by the same factor horizontally does not change itssize. Moving it appropriately, the area fits the graph of thefunctionf(x) = 1/xagain. Therefore, the left hand bluearea, which is the integral of f(x) fromttotuis the same

as the integral from 1 tou. This justifies the equality (2)with a more geometric proof.

tu tuu

ln(u)

u

ln(tu)

ln(t)

A visual proof of the product formula of the natural logarithm

The power formula ln(tr) = rln(t) may be derived in asimilar way:

ln(tr) =

tr1

1

xdx=

t1

1

wr

rwr1 dw

= r

t1

1

wdw = r ln(t).

The second equality uses a change of variables(integration by substitution),w=x1/r.

The sum over the reciprocals of natural numbers,

1 +12

+13

+ + 1n

=

nk=1

1k

,

is called the harmonic series. It is closely tied to thenatural logarithm: asntends toinfinity, the difference,

nk=1

1

k ln(n),

converges(i.e., gets arbitrarily close) to a number knownas the EulerMascheroni constant. This relation aids

in analyzing the performance of algorithms such asquicksort.[50]

There is also another integral representation of the loga-rithm that is useful in some situations.

ln(x) = lim0

dt

t

ext et

This can be verified by showing that it has the same valueatx= 1, and the same derivative.

5.5 Transcendence of the logarithm

Real numbers that are not algebraic are calledtranscendental;[51] for example, and e are such

numbers, but

2

3is not. Almost allreal numbersare transcendental. The logarithm is an example of atranscendental function. The GelfondSchneider theo-remasserts that logarithms usually take transcendental,i.e., difficult values.[52]

6 Calculation

Logarithms are easy to compute in some cases, such aslog10(1,000) = 3. In general, logarithms can be calculatedusingpower seriesor thearithmetic-geometric mean, orbe retrieved from a precalculated logarithm table thatprovides a fixed precision.[53][54] Newtons method, aniterative method to solve equations approximately, canalso be used to calculate the logarithm, because its in-verse function, the exponential function, can be com-puted efficiently.[55] Using look-up tables,CORDIC-likemethods can be used to compute logarithms if the onlyavailable operations are addition and bit shifts.[56][57]

Moreover, the binary logarithm algorithm calculates lb(x)recursivelybased on repeated squarings of x, taking ad-vantage of the relation
https://en.wikipedia.org/wiki/Recursionhttps://en.wikipedia.org/wiki/Recursionhttps://en.wikipedia.org/wiki/Binary_logarithm#Algorithmhttps://en.wikipedia.org/wiki/Arithmetic_shifthttps://en.wikipedia.org/wiki/CORDIChttps://en.wikipedia.org/wiki/Newton%27s_methodhttps://en.wikipedia.org/wiki/Logarithm_tablehttps://en.wikipedia.org/wiki/Arithmetic-geometric_meanhttps://en.wikipedia.org/wiki/Power_serieshttp://localhost/var/www/apps/conversion/tmp/scratch_1/%EE%BF%80http://localhost/var/www/apps/conversion/tmp/scratch_1/%EE%BF%80https://en.wikipedia.org/wiki/Transcendental_functionhttps://en.wikipedia.org/wiki/Almost_allhttps://en.wikipedia.org/wiki/E_(mathematical_constant)https://en.wikipedia.org/wiki/Pihttps://en.wikipedia.org/wiki/Transcendental_numberhttps://en.wikipedia.org/wiki/Algebraic_numberhttps://en.wikipedia.org/wiki/Real_numberhttps://en.wikipedia.org/wiki/Quicksorthttp://localhost/var/www/apps/conversion/tmp/scratch_1/%EE%BF%80https://en.wikipedia.org/wiki/Limit_of_a_sequencehttps://en.wikipedia.org/wiki/Infinityhttps://en.wikipedia.org/wiki/Natural_logarithmhttps://en.wikipedia.org/wiki/Harmonic_series_(mathematics)https://en.wikipedia.org/wiki/Integration_by_substitutionhttps://en.wikipedia.org/wiki/Natural_logarithmhttps://en.wikipedia.org/wiki/Fundamental_theorem_of_calculushttps://en.wikipedia.org/wiki/Integralhttps://en.wikipedia.org/wiki/Natural_logarithmhttps://en.wikipedia.org/wiki/Natural_logarithmhttps://en.wikipedia.org/wiki/Natural_logarithmhttps://en.wikipedia.org/wiki/Natural_logarithm


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8 6 CALCULATION

log2(x2) = 2 log2(x).

6.1 Power series

Taylor series

TheTaylor seriesof ln(z) centered atz = 1. Theanimation showsthe 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 satisfies 0 0.[nb 5][58] Using theSigma nota-tion, this is also written as

ln(z) = 2n=0

1

2n + 1

z 1z+ 1

2n+1.

This series can be derived from the above Taylor series. It

converges more quickly than the Taylor series, especiallyifzis close to 1. For example, for z= 1.5, the first threeterms of the second series approximate ln(1.5) with anerror of about 3106. The quick convergence forzcloseto 1 can be taken advantage of in the following way: givena low-accuracy approximationy ln(z) and putting

A= z

exp(y),

the logarithm ofzis:

ln(z) =y +ln(A).

The better the initial approximationyis, the closerAis to1, so its logarithm can be calculated efficiently.Acan becalculated using theexponential series, which convergesquickly providedyis not too large. Calculating the loga-rithm of largerzcan be reduced to smaller values ofzbywritingz=a 10b, so that ln(z) = ln(a) + b ln(10).

A closely related method can be used to compute the log-arithm of integers. From the above series, it follows that:

ln(n + 1) =ln(n) + 2k=0

1

2k+ 1

1

2n + 1

2k+1.

If the logarithm of a large integer n is known, then thisseries yields a fast converging series for log(n+1).

6.2 Arithmetic-geometric mean approxi-

mation

Thearithmetic-geometric meanyields high precision ap-

proximations of thenatural logarithm. ln(x) is approxi-mated to a precision of 2p (orpprecise bits) by the fol-lowing formula (due toCarl Friedrich Gauss):[59][60]
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7.1 Logarithmic scale 9

ln(x) 2M(1, 22m/x)

m ln(2).

HereMdenotes the arithmetic-geometric mean. It is ob-tained by repeatedly calculating the average (arithmeticmean) and the square root of the product of two numbers(geometric mean). Moreover,mis chosen such that

x 2m >2p/2.

Both the arithmetic-geometric mean and the constants and ln(2) can be calculated with quickly converging se-ries.

7 Applications

A nautilus displaying a logarithmic spiral

Logarithms have many applications inside and outsidemathematics. Some of these occurrences are relatedto the notion of scale invariance. For example, eachchamber of the shell of a nautilus is an approximatecopy of the next one, scaled by a constant factor. Thisgives rise to alogarithmic spiral.[61] Benfords lawon thedistribution of leading digits can also be explained byscale invariance.[62] Logarithms are also linked to self-similarity. For example, logarithms appear in the analysisof algorithms that solve a problem by dividing it into twosimilar smaller problems and patching their solutions.[63]

The dimensions of self-similar geometric shapes, that is,shapes whose parts resemble the overall picture are alsobased on logarithms. Logarithmic scalesare useful forquantifying the relative change of a value as opposed toits absolute difference. Moreover, because the logarith-mic function log(x) grows very slowly for largex, logarith-mic scales are used to compress large-scale scientific data.

Logarithms also occur in numerous scientific formulas,such as the Tsiolkovsky rocket equation, the Fenske equa-tion, or theNernst equation.

7.1 Logarithmic scale

Main article:Logarithmic scaleScientific quantities are often expressed as logarithms of

A logarithmic chart depicting the value of one Goldmark in

Papiermarksduring theGerman hyperinflation in the 1920s

other quantities, using a logarithmic scale. For example,thedecibelis a logarithmic unit of measurement. It isbased on the common logarithm ofratios10 times thecommon logarithm of apowerratio or 20 times the com-mon logarithm of a voltage ratio. It is used to quantify theloss of voltage levels in transmitting electrical signals,[64]

to describe power levels of sounds in acoustics,[65] andtheabsorbanceof light in the fields ofspectrometryandoptics. Thesignal-to-noise ratiodescribing the amountof unwantednoisein relation to a (meaningful)signalisalso measured in decibels.[66] In a similar vein, thepeaksignal-to-noise ratiois commonly used to assess the qual-

ity of sound and image compressionmethods using thelogarithm.[67]

The strength of an earthquake is measured by taking thecommon logarithm of the energy emitted at the quake.This is used in the moment magnitude scale or the Richterscale. For example, a 5.0 earthquake releases 10 timesand a 6.0 releases 100 times the energy of a 4.0.[68] An-other logarithmic scale isapparent magnitude. It mea-sures the brightness of stars logarithmically.[69] Yet an-other example ispHinchemistry; pH is the negative ofthe common logarithm of theactivityofhydroniumions(the formhydrogen ionsH+

take in water).[70] The activity of hydronium ions in neu-tral water is 107 molL1, hence a pH of 7. Vinegar typ-ically has a pH of about 3. The difference of 4 corre-
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10 7 APPLICATIONS

sponds to a ratio of 104 of the activity, that is, vinegarshydronium ion activity is about 103 molL1.

Semilog (log-linear) graphs use the logarithmic scale con-cept 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 1trillion to the same space (on the vertical axis) as the in-crease from 1 to 1 million. In such graphs,exponentialfunctionsof the formf(x) =abx appear as straight lineswithslopeequal to the logarithm of b. Log-loggraphsscale both axes logarithmically, which causes functionsof the form f(x) = a xk to be depicted as straight lineswith slope equal to the exponentk. This is applied in vi-sualizing and analyzingpower laws.[71]

7.2 Psychology

Logarithms occur in several laws describinghuman per-ception:[72][73] Hicks lawproposes a logarithmic relationbetween the time individuals take for choosing an alter-native and the number of choices they have.[74] Fittss lawpredicts that the time required to rapidly move to a tar-get area is a logarithmic function of the distance to andthe size of the target.[75] Inpsychophysics, theWeberFechner lawproposes a logarithmic relationship betweenstimulus and sensation such as the actual vs. the perceivedweight of an item a person is carrying.[76] (This law,however, is less precise than more recent models, such astheStevens power law.[77])

Psychological studies found that individuals with littlemathematics education tend to estimate quantities log-arithmically, that is, they position a number on an un-marked line according to its logarithm, so that 10 is po-sitioned as close to 100 as 100 is to 1000. Increasing ed-ucation shifts this to a linear estimate (positioning 100010x as far away) in some circumstances, while logarithmsare used when the numbers to be plotted are difficult toplot linearly.[78][79]

7.3 Probability theory and statistics

Logarithms arise inprobability theory: thelaw of largenumbersdictates that, for afair coin, as the number ofcoin-tosses increases to infinity, the observed proportionof headsapproaches one-half. The fluctuations of thisproportion about one-half are described by thelaw of theiterated logarithm.[80]

Logarithms also occur inlog-normal distributions. Whenthe logarithm of a random variable has a normal dis-tribution, the variable is said to have a log-normaldistribution.[81] Log-normal distributions are encoun-tered in many fields, wherever a variable is formed as the

product of many independent positive random variables,for example in the study of turbulence.[82]

Logarithms are used formaximum-likelihood estimation

Threeprobability density functions(PDF) of random variables

with log-normal distributions. The location parameter, which iszero 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 countriesof the world. Black dots indicate the distribution

predicted by Benfords law.

of parametricstatistical models. For such a model, thelikelihood function depends on at least one parameter that

must be estimated. A maximum of the likelihood func-tion occurs at the same parameter-value as a maximumof 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 multi-plied likelihoods forindependentrandom variables.[83]

Benfords lawdescribes the occurrence of digits in manydata sets, such as heights of buildings. According to Ben-fords law, the probability that the first decimal-digit of anitem in the data sample is d(from 1 to 9) equals log10(d+1) log10(d), regardlessof the unit of measurement.

[84]

Thus, about 30% of the data can be expected to have

1 as first digit, 18% start with 2, etc. Auditors exam-ine deviations from Benfords law to detect fraudulentaccounting.[85]
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7.6 Fractals 11

7.4 Computational complexity

Analysis of algorithmsis a branch of computer sciencethat studies the performance of algorithms (computerprograms solving a certain problem).[86] Logarithms arevaluable for describing algorithms that divide a prob-

lem into smaller ones, and join the solutions of thesubproblems.[87]

For example, to find a number in a sorted list, the binarysearch algorithmchecks the middle entry and proceedswith the half before or after the middle entry if the num-ber is still not found. This algorithm requires, on average,log2(N) comparisons, where Nis thelists length.

[88] Sim-ilarly, themerge sortalgorithm sorts an unsorted list bydividing the list into halves and sorting these first beforemerging the results. Merge sort algorithms typically re-quire a timeapproximately proportional toN log(N).[89]

The base of the logarithm is not specified here, because

the result only changes by a constant factor when anotherbase is used. A constant factor, is usually disregardedin the analysis of algorithms under the standarduniformcost model.[90]

A functionf(x) is said togrow logarithmicallyif f(x) is(exactly or approximately) proportional to the logarithmofx. (Biological descriptions of organism growth, how-ever, use this term for an exponential function.[91]) Forexample, anynatural number Ncan be represented inbinary formin no more than log2(N) + 1bits. In otherwords, the amount ofmemoryneeded to store Ngrowslogarithmically withN.

7.5 Entropy and chaos

Billiardson an ovalbilliard table. Two particles, starting at the

center with an angle differing by one degree, take paths that di-

verge chaotically because ofreflectionsat the boundary.

Entropyis broadly a measure of the disorder of some sys-tem. In statistical thermodynamics, the entropy Sof somephysical system is defined as

S= k

ipiln(pi).

The sum is over all possible statesiof the system in ques-tion, such as the positions of gas particles in a container.

Moreover,piis the probability that the stateiis attainedand kis theBoltzmann constant. Similarly,entropy ininformation theorymeasures the quantity of information.If a message recipient may expect any one ofNpossiblemessages with equal likelihood, then the amount of infor-mation conveyed by any one such message is quantified

as log2(N) bits.[92]

Lyapunov exponentsuse logarithms to gauge the degreeof chaoticity of a dynamical system. For example, fora particle moving on an oval billiard table, even smallchanges of the initial conditions result in very differ-ent paths of the particle. Such systems arechaoticin adeterministicway, because small measurement errors ofthe initial state predictably lead to largely different finalstates.[93] At least one Lyapunov exponent of a determin-istically chaotic system is positive.

7.6 Fractals

The Sierpinski triangle (at the right) is constructed by repeatedly

replacingequilateral trianglesby three smaller ones.

Logarithms occur in definitions of the dimension offractals.[94] Fractals are geometric objects that are self-similar: small parts reproduce, at least roughly, the entireglobal structure. The Sierpinski triangle (pictured) can be

covered by three copies of itself, each having sides halfthe original length. This makes theHausdorff dimensionof this structure log(3)/log(2) 1.58. Another logarithm-based notion of dimension is obtained by counting thenumber of boxesneeded to cover the fractal in question.

7.7 Music

Four different octaves shown on a linear scale, thenshown on a logarithmic scale (as the ear hears them).

Logarithms are related to musical tones andintervals. Inequal temperament, the frequency ratio depends only onthe interval between two tones, not on the specific fre-quency, orpitch, of the individual tones. For example,thenoteAhas a frequency of 440HzandB-flathas a fre-

quency of 466 Hz. The interval betweenA and B-flatisasemitone, as is the one betweenB-flatandB(frequency493 Hz). Accordingly, the frequency ratios agree:
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12 8 GENERALIZATIONS

466

440 493

466 1.059 12

2.

Therefore, logarithms can be used to describe the in-tervals: an interval is measured in semitones by taking

the base-21/12

logarithm of thefrequencyratio, while thebase-21/1200 logarithm of the frequency ratio expressesthe interval incents, hundredths of a semitone. The lat-ter is used for finer encoding, as it is needed for non-equaltemperaments.[95]

7.8 Number theory

Natural logarithmsare closely linked tocounting primenumbers(2, 3, 5, 7, 11, ...), an important topic innumbertheory. For anyintegerx, the quantity ofprime numbersless than or equal toxis denoted(x). The prime number

theoremasserts that (x) is approximately given by

x

ln(x),

in the sense that the ratio of (x) and that fraction ap-proaches 1 whenxtends to infinity.[96] As a consequence,the probability that a randomly chosen number between 1andxis prime is inverselyproportionalto the numbers ofdecimal digits ofx. A far better estimate of (x) is givenby theoffset logarithmic integralfunction Li(x), definedby

Li(x) =

x2

1

ln(t)dt.

TheRiemann hypothesis, one of the oldest open math-ematicalconjectures, can be stated in terms of compar-ing (x) and Li(x).[97] TheErdsKac theoremdescrib-ing the number of distinctprime factorsalso involves thenatural logarithm.

The logarithm ofnfactorial,n! = 1 2 ... n, is given by

ln(n!) =ln(1) +ln(2) + +ln(n).This can be used to obtainStirlings formula, an approx-imation ofn! for largen.[98]

8 Generalizations

8.1 Complex logarithm

Main article:Complex logarithmThecomplex numbersasolving the equation

ea =z.

Polar form ofz = x + iy. Both and ' are arguments ofz.

are calledcomplex logarithms. Here,zis a complex num-ber. A complex number is commonly represented asz = x+ iy, wherexand y are real numbers and iis the imaginaryunit. Such a number can be visualized by a point in thecomplex plane, as shown at the right. Thepolar formen-codes a non-zero complex numberzby itsabsolute value,that is, the distancerto theorigin, and an angle betweenthexaxis and the line passing through the origin and z.This angle is called theargumentofz. The absolute valuerofzis

r=

x2 + y2.

The argument is not uniquely specified by z: both and'=+2areargumentsofz because adding 2 radiansor 360 degrees[nb 6] to corresponds to winding aroundthe origin counter-clock-wise by aturn. The resultingcomplex number is again z, as illustrated at the right.However, exactly one argument satisfies < and . It is called the principal argument, denoted Arg(z),with a capitalA.[99] (An alternative normalization is 0

Arg(z) < 2.[100])

Using trigonometric functions sine and cosine, or thecomplex exponential, respectively, rand are such thatthe following identities hold:[101]

z = r (cos + i sin )= rei.

This implies that thea-th power ofeequalsz, where

a= ln(r) + i( + 2n),

is the principal argument Arg(z) and n is an arbitraryinteger. Any such a is called a complex logarithm of
https://en.wikipedia.org/wiki/Complex_exponentialhttps://en.wikipedia.org/wiki/Cosinehttps://en.wikipedia.org/wiki/Sinehttps://en.wikipedia.org/wiki/Trigonometric_functionshttps://en.wikipedia.org/wiki/Turn_(geometry)https://en.wikipedia.org/wiki/Radianhttps://en.wikipedia.org/wiki/Argument_(complex_analysis)https://en.wikipedia.org/wiki/Origin_(mathematics)https://en.wikipedia.org/wiki/Absolute_valuehttps://en.wikipedia.org/wiki/Polar_formhttps://en.wikipedia.org/wiki/Complex_planehttps://en.wikipedia.org/wiki/Imaginary_unithttps://en.wikipedia.org/wiki/Imaginary_unithttps://en.wikipedia.org/wiki/Complex_numberhttps://en.wikipedia.org/wiki/Complex_logarithmhttps://en.wikipedia.org/wiki/Stirling%27s_formulahttps://en.wikipedia.org/wiki/Factorialhttps://en.wikipedia.org/wiki/Natural_logarithmhttps://en.wikipedia.org/wiki/Prime_factorhttp://localhost/var/www/apps/conversion/tmp/scratch_1/%EE%BF%80https://en.wikipedia.org/wiki/Conjecturehttps://en.wikipedia.org/wiki/Riemann_hypothesishttps://en.wikipedia.org/wiki/Logarithmic_integral_functionhttps://en.wikipedia.org/wiki/Proportionality_(mathematics)https://en.wikipedia.org/wiki/Prime_number_theoremhttps://en.wikipedia.org/wiki/Prime_number_theoremhttps://en.wikipedia.org/wiki/Prime-counting_functionhttps://en.wikipedia.org/wiki/Prime_numberhttps://en.wikipedia.org/wiki/Integerhttps://en.wikipedia.org/wiki/Number_theoryhttps://en.wikipedia.org/wiki/Number_theoryhttps://en.wikipedia.org/wiki/Prime-counting_functionhttps://en.wikipedia.org/wiki/Prime-counting_functionhttps://en.wikipedia.org/wiki/Natural_logarithmhttps://en.wikipedia.org/wiki/Cent_(music)https://en.wikipedia.org/wiki/Frequency


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8.3 Related concepts 13

The principal branch of the complex logarithm, Log(z). Theblack point at z = 1 corresponds to absolute value zero andbrighter (moresaturated) colors refer to bigger absolute values.

Thehueof the color encodes the argument of Log(z).

z. There are infinitely many of them, in contrast to theuniquely defined real logarithm. If n = 0,a is called the

principal value of the logarithm, denoted Log(z). Theprincipal argument of any positive real number x is 0;hence Log(x) is a real number and equals the real (nat-ural) logarithm. However, the above formulas for loga-rithms of products and powersdo not generalizeto the

principal value of the complex logarithm.[102]

The illustration at the right depicts Log(z). The discon-tinuity, that is, the jump in the hue at the negative partof thex- or real axis, is caused by the jump of the prin-cipal argument there. This locus is called abranch cut.This behavior can only be circumvented by dropping therange restriction on . Then the argument ofzand, con-sequently, its logarithm becomemulti-valued functions.

8.2 Inverses of other exponential functions

Exponentiation occurs in many areas of mathematics andits inverse function is often referred to as the logarithm.For example, the logarithm of a matrix is the (multi-valued) inverse function of thematrix exponential.[103]

Another example is the p-adic logarithm, the inversefunction of the p-adic exponential. Both are defined viaTaylor series analogous to the real case.[104] In the contextofdifferential geometry, the exponential mapmaps thetangent spaceat a point of amanifoldto a neighborhoodof that point. Its inverse is also called the logarithmic (orlog) map.[105]

In the context offinite groupsexponentiation is given by

repeatedly multiplying one group element b with itself.Thediscrete logarithmis the integer n solving the equa-tion

bn =x,

where x is an element of the group. Carrying out theexponentiation can be done efficiently, but the discrete

logarithm is believed to be very hard to calculate insome groups. This asymmetry has important applicationsinpublic key cryptography, such as for example in theDiffieHellman key exchange, a routine that allows se-cure exchanges ofcryptographickeys over unsecured in-formation channels.[106] Zechs logarithmis related to thediscrete logarithm in the multiplicative group of non-zeroelements of afinite field.[107]

Further logarithm-like inverse functions include the dou-ble logarithm ln(ln(x)), the super- or hyper-4-logarithm(a slight variation of which is called iterated logarithmin computer science), theLambert W function, and the

logit. They are the inverse functions of thedouble expo-nential function,tetration, of f(w) = wew,[108] and of thelogistic function, respectively.[109]

8.3 Related concepts

From the perspective ofpure mathematics, the identitylog(cd) = log(c) + log(d) expresses agroup isomorphismbetween positiverealsunder multiplication and reals un-der addition. Logarithmic functions are the only continu-ous isomorphisms between these groups.[110] By means ofthat isomorphism, theHaar measure(Lebesgue measure)dxon the reals corresponds to the Haar measure dx/xonthe positive reals.[111] Incomplex analysisandalgebraicgeometry,differential formsof the form df/fare knownas forms with logarithmicpoles.[112]

Thepolylogarithmis the function defined by

Lis(z) =k=1

zk

ks.

It is related to thenatural logarithmby Li1(z) = ln(1

z). Moreover, Lis(1) equals theRiemann zeta function(s).[113]

9 See also

Exponential function

Index of logarithm articles

10 Notes

[1] The restrictions on xand bare explained in the sectionAnalytic properties.
https://en.wikipedia.org/wiki/Logarithm#Analytic_propertieshttps://en.wikipedia.org/wiki/Index_of_logarithm_articleshttps://en.wikipedia.org/wiki/Exponential_functionhttps://en.wikipedia.org/wiki/Riemann_zeta_functionhttps://en.wikipedia.org/wiki/Natural_logarithmhttps://en.wikipedia.org/wiki/Polylogarithmhttps://en.wikipedia.org/wiki/Pole_(complex_analysis)https://en.wikipedia.org/wiki/Differential_formhttps://en.wikipedia.org/wiki/Algebraic_geometryhttps://en.wikipedia.org/wiki/Algebraic_geometryhttps://en.wikipedia.org/wiki/Complex_analysishttps://en.wikipedia.org/wiki/Lebesgue_measurehttps://en.wikipedia.org/wiki/Haar_measurehttps://en.wikipedia.org/wiki/Real_numberhttps://en.wikipedia.org/wiki/Group_isomorphismhttps://en.wikipedia.org/wiki/Pure_mathematicshttps://en.wikipedia.org/wiki/Logistic_functionhttps://en.wikipedia.org/wiki/Tetrationhttps://en.wikipedia.org/wiki/Double_exponential_functionhttps://en.wikipedia.org/wiki/Double_exponential_functionhttps://en.wikipedia.org/wiki/Logithttps://en.wikipedia.org/wiki/Lambert_W_functionhttps://en.wikipedia.org/wiki/Iterated_logarithmhttps://en.wikipedia.org/wiki/Super-logarithmhttps://en.wikipedia.org/wiki/Finite_fieldhttps://en.wikipedia.org/wiki/Zech%27s_logarithmhttps://en.wikipedia.org/wiki/Cryptographyhttp://localhost/var/www/apps/conversion/tmp/scratch_1/%EE%BF%80https://en.wikipedia.org/wiki/Public_key_cryptographyhttps://en.wikipedia.org/wiki/Discrete_logarithmhttps://en.wikipedia.org/wiki/Finite_groupshttps://en.wikipedia.org/wiki/Neighborhood_(mathematics)https://en.wikipedia.org/wiki/Differentiable_manifoldhttps://en.wikipedia.org/wiki/Tangent_spacehttps://en.wikipedia.org/wiki/Exponential_maphttps://en.wikipedia.org/wiki/Differential_geometryhttps://en.wikipedia.org/wiki/P-adic_exponential_functionhttps://en.wikipedia.org/wiki/P-adic_logarithm_functionhttps://en.wikipedia.org/wiki/Matrix_exponentialhttps://en.wikipedia.org/wiki/Logarithm_of_a_matrixhttps://en.wikipedia.org/wiki/Multi-valued_functionhttps://en.wikipedia.org/wiki/Branch_cuthttps://en.wikipedia.org/wiki/Exponentiation#Failure_of_power_and_logarithm_identitieshttps://en.wikipedia.org/wiki/Huehttps://en.wikipedia.org/wiki/Saturation_(color_theory)


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14 11 REFERENCES

[2] Some mathematicians disapprove of this notation. In his1985 autobiography,Paul Halmoscriticized what he con-sidered the childish ln notation, which he said no math-ematician had ever used.[12] The notation was invented byIrving Stringham, a mathematician.[13][14]

[3] For exampleC,Java,Haskell, andBASIC.

[4] The same series holds for the principal value of the com-plex logarithm for complex numberszsatisfying |z 1|

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