LogarithmFrom Wikipedia, the free encyclopedia

Logarithm functions, graphed for various bases: red is to base e, green is to base 10, and purple is to base 1.7. Each tick on the axes is one unit. Logarithms

of all bases pass through the point (1, 0), because any non-zero number raised to the power 0 is 1, and through the points (b, 1) for base b, because a

number raised to the power 1 is itself. The curves approach the y-axis but do not reach it because of the singularity at x = 0 (a vertical asymptote).

The 1797 Encyclopædia Britannica explains logarithms as "a series of numbers in arithmetical progression, corresponding to others in geometrical

progression; by means of which, arithmetical calculations can be made with much more ease and expedition than otherwise."

In mathematics, the logarithm of a number to a given base is the power or exponent to which the base must be raised in order to produce

that number. For example, the logarithm of 1000 to base 10 is 3, because 3 is the power to which ten must be raised to produce 1000:

103 = 1000, so log101000  = 3. Only positive real numbers have real number logarithms; negative and complex numbers have complex

logarithms.

The logarithm of x to the base b is written logb(x) or, if the base is implicit, as log(x). So, for a number x, a base b and an exponent y,

The bases used most often are 10 for the common logarithm, e for the natural logarithm, and 2 for the binary logarithm.

An important feature of logarithms is that they reduce multiplication to addition, by the formula:

That is, the logarithm of the product of two numbers is the sum of the logarithms of those numbers.

Similarly, logarithms reduce division to subtraction by the formula:

That is, the logarithm of the quotient of two numbers is the difference between the logarithms of those numbers.

The use of logarithms to facilitate complicated calculations was a significant motivation in their original development. Logarithms have applications in fields as diverse as statistics, chemistry, physics, astronomy, computer science, economics, music, and engineering.

ogarithm of positive real numbers

[Definition

The graph of the function f(x) = 2x (red) together with a depiction of log2(3) ≈ 1.58.

The logarithm of a positive real number y with respect to another positive real number b, where b is not equal to 1, is the real

number x such that

that is, the x-th power of b must equal y.[1][2]

The logarithm x is denoted logb(y). (Some European countries write blog(y) instead.[3]) The number b is referred to as the base.

For b = 2, for example, this means

since 23 = 2 · 2 · 2 = 8. The logarithm may be negative, for example

since

The right image shows how to determine (approximately) the logarithm. Given the graph (in red) of the function f(x) = 2x, the logarithm

log2(y) is the For any given number y (y = 3 in the image), the logarithm of y to the base 2 is the x-coordinate of the intersection point of

the graph and the horizontal line intersecting the vertical axis at 3.

Above, the logarithm has been defined to be the solution of an equation. For this to be meaningful, it is thus necessary to ensure that

there is always exactly one such solution. This is done using three properties of the function f(x) = bx: in the case b > 1, this function f(x)

is strictly increasing, that is to say, f(x) increases when x does so. Secondly, the function takes arbitrarily big values and arbitrarily small

positive values. Thirdly, the function is continuous. Intuitively, the function does not "jump": the graph can be drawn without lifting the

pen. These properties, together with the intermediate value theorem ofelementary calculus ensure that there is indeed exactly one

solution x to the equation

f(x) = bx = y,

for any given positive y. When 0 < b < 1, a similar argument is used, except that f(x) = bx is decreasing in that case.

[edit]Identities

Main article: Logarithmic Identities

The above definition of the logarithm implies a number of properties.

[edit]Logarithm of products

Logarithms map multiplication to addition. That is to say, for any two positive real numbers x and y, and a given positive base b, the

identity

logb(x · y) = logb(x) + logb(y).

For example,

log3(9 · 27) = log3(243) = 5,

since 35 = 243. On the other hand, the sum of log3(9) = 2 and log3(27) = 3 also equals 5. In general, that identity is derived from the

relation of powers and multiplication:

bs · bt = bs + t.

Indeed, with the particular values s = logb(x) and t = logb(y), the preceding equality implies

logb(bs · bt) = logb(bs + t) = s + t = logb(bs) + logb(bt).

By virtue of this identity, logarithms make lengthy numerical operations easier to perform by converting multiplications to additions. The

manual computation process is made easy by using tables of logarithms, or a slide rule. The property of common logarithms pertinent to the

use of log tables is that any decimal sequence of the same digits, but different decimal-point positions, will have identical mantissas and

differ only in their characteristics.

Logarithm of powers

A related property is reduction of exponentiation to multiplication. Another way of rephrasing the definition of the logarithm is to write

x = blogb(x).

Raising both sides of the equation to the p-th power (exponentiation) shows

xp = (blogb(x))p = bp · log

b(x).

thus, by taking logarithms:

logb(xp) = p logb(x).

In prose, the logarithm of the p-th power of x is p times the logarithm of x. As an example,

log2(64) = log2(43) = 3 · log2(4) = 3 · 2 = 6.

Besides reducing multiplication operations to addition, and exponentiation to multiplication, logarithms reduce division to subtraction,

and roots to division. For example,

Change of base

The above rule for a logarithm of a power can be used to derive a relation between logarithms with respect to different bases:

In fact, the left hand side of the above is the unique number a such that ba = x. Therefore

logk(x) = logk(ba) = a · logk(b).

The general restriction b ≠ 1 implies logkb ≠ 0, since b0 = 1. Thus, dividing the preceding equation by logkb shows the above

formula.

As a practical consequence, logarithms with respect to any base k can be calculated, e.g. using a calculator, if logarithms

to the base b are available. From a more theoretical viewpoint, this result implies that the graphs of all logarithm functions

(whatever the base b) are similar to each other.

One way of viewing the change-of-base formula is to say that in the expression

the two bs "cancel", leaving logk x.

[edit]Bases

While the definition of logarithm applies to any positive real number b (1 is excluded, though), a few particular choices for b are

more commonly used. These are b = 10, b = e (the mathematical constant ≈ 2.71828…), and b = 2. The different standards

come about because of the different properties preferred in different fields.

For b = 10, the logarithm log10 is called common logarithm. It appears in various engineering fields, especially for

power levels and power ratios, such as acoustical sound pressure, and in logarithm tables to be used to simplify

hand calculations. Its use is historically grounded (see dB)[citation needed]. The logarithm of a number in a given base can show

how many digits needed to write that number in that base. For instance, the common logarithm of a number x tells how

many numerical digits x has: when

n − 1 ≤ log10(x) < n,

with an integer n then x has n decimal digits. Since we write numbers in base 10, mental math is thus easier with

the common log[citation needed], making it attractive to many engineers. The approximation 210 ≈ 103 leads to the approximations

3 dB per octave (power doubling) – a useful result that occurs with the use of log10.

The natural logarithm, loge(x) is the one with base b = e. It has many "natural" properties related to

its analytical behavior explained below. It is found in mathematical analysis, statistics, economics and some

engineering fields. For example, Euler's identity is important to fields that deal with cyclic components. The natural

logarithm of x is often written "ln(x)", instead of loge(x) especially in disciplines where it isn't written "log(x)". However, some

mathematicians disapprove of this notation. In his 1985 autobiography, Paul Halmos criticized what he considered the

"childish ln notation," which he said no mathematician had ever used.[4] In fact, the notation was invented by a

mathematician, Irving Stringham, professor of mathematics at University of California, Berkeley, in

1893.[5][6]

The binary logarithm with base b = 2 is used computer science and information theory. Computers

ubiquitously use binary storage with bits as the basic unit and it takes at least ⌊log2(n)⌋+1 bits to store the integer n. Likewise,

a binary search through a sorted list of size n takes ⌊log2(n)⌋+1 steps. Properties like this come up repeatedly in these

domains.

[edit]Implicit bases

Instead of writing logb(x), it is common to omit the base, log(x), when intended base can be determined from context. In

mathematics and many programming languages,[7] "log(x)" is usually understood to be the natural logarithm.

Engineers, biologists and astronomers often define "log(x)" to be the common logarithm, log10(x), while computer scientists

often choose "log(x)" to be the binary logarithm, log2(x).

On most calculators, the "log" button is log10(x) and "ln" is loge(x). The International Organization for Standardization (ISO 31-11) suggests the notations "ln(x)", "lg(x)", "lb(x)" for loge(x), log10(x), and log2(x),

respectively.[8]

The base, b, used by the supplied logarithm function can be explicitly determined using the following identity (subject to the

inherent computational accuracy errors).

This follows from the change-of-base formula above.

[edit]Computer science

In computer science, the base-2 logarithm is sometimes written as lg(x), as suggested by Edward Reingold and

popularized by Donald Knuth. However, lg(x) is also sometimes used for the common logarithm, and lb(x) for the base-2

logarithm.[9] In Russian literature, the notation lg(x) is also generally used for the base-10 logarithm.[10] In German, lg(x) also

denotes the base-10 logarithm, while sometimes ld(x) or lb(x) is used for the base-2 logarithm. The PL/I Programming language

uses log2(x) for the base-2 logarithm.

[edit]Equivalence of logarithms

The above identity relating logarithms with respect to different bases shows that the difference between logarithms to different

bases is one of scale. For example, the unit decibel (dB) refers to a common logarithm (b = 10). Alternatively, neper are

based on a natural logarithm. Another example from information theory: calculations carried out using log2 will lead to

results in bits, which has an intuitive meaning; corresponding calculations carried out using loge will lead to results

in nats which may lack this intuitive interpretation. However, the change amounts to a factor of loge2≈0.69—twice as many

values can be encoded with one additional bit, which corresponds to an increase of about 0.69 nats.

Logarithms to a base less than 1 have a negative scale, or a flip about the x axis, relative to logarithms of base greater than 1.

In disciplines where the scale is irrelevant, the term indefinite logarithm refers to point of view. An example

is complexity theory which describes the asymptotic behavior of algorithms in big O notation. It often

makes statements like "the behavior of the algorithm is logarithmic", but does not measure of performance of the algorithm in a

given situation.

Logarithm as a function

The graph of the logarithm function logb(x) (green) is obtained by reflecting the one of the function bx (red) at the diagonal line (x = y).

The expression logb(x) depends on both b and x, but the term logarithm function (or logarithmic function) refers to a function of

the form logb(x) in which the base bis fixed and x is variable, thus yielding a function that assigns to any x its logarithm logb(x).

The word "logarithm" is often used to refer to a logarithm function itself as well as to particular values of this function. The

above definition of logarithms were done indirectly by means of the exponential function. A compact way of rephrasing that

definition is to say that the base-b logarithm function is the inverse function of the exponential function bx: a point (t, u = b(t)) on

the graph of the exponential function yields a point (u, t = logbu) on the graph of the logarithm and vice versa. Geometrically,

this corresponds to the statement that the points correspond one to another upon reflecting them at the diagonal line x = y.

Using this relation to the exponential function, calculus tells that the logarithm function is continuous (it does not "jump", i.e., the

logarithm of x changes only little when x varies only little). What is more, it is differentiable (intuitively, this means that the graph

of logb(x) has no sharp "corners").

[edit]Integral representation of the natural logarithm

The natural logarithm of t is the shaded area underneath the graph of the functionf(x) = 1/x (reciprocal of x).

When b = e (Euler's number), the natural logarithm ln(t) = loge(t) can be shown to satisfy the following identity:

In prose, the natural logarithm of t agrees with the integral of 1/x dx from 1 to t, that is to say, the area between the x-

axis and the function 1/x, ranging from x = 0 tox = t. This is depicted at the right. Some authors use the right hand side of

this equation as a definition of the natural logarithm.[citation needed] The formulae of logarithms of products and powers

mentioned above can be derived directly from this presentation of the natural logarithm. The product formula ln(tu) = ln(t)

+ ln(u) is deduced in the following way:

The "=" labelled (1) used a splitting of the integral into two parts, the equality (2) is a change of variable (w = x/t).

This can also be understood geometrically. The integral is split into two parts (shown in yellow and blue). The key

point is this: rescaling the left hand blue area in vertical direction by the factor t and shrinking it by the same factor

in the horizontal direction does not change its size. Moving it appropriately, the area fits the graph of the function

1/x again. Therefore, the left hand area, which is the integral of f(x) from t to tu is the same as the integral

from 1 to u:

A visual proof of the product formula of the natural logarithm.

The power formula ln(tr) = r ln(t) is derived similarly:

The second equality is using a change of variables, w := x1/r, while the third equality follows from integration

by substitution.

[edit]Derivative and antiderivativeFurther information: List of integrals of logarithmic functions

The fundamental theorem of calculus applied to the above integral formula shows that the derivative of the

natural logarithm function is

More generally, by the chain rule, the derivative with a generalised functional argument f(x) is

For this reason the quotient at right hand side is called logarithmic derivative of f.

The antiderivative of the natural logarithm ln(x) is

Calculation

[edit]Taylor series

There are several series for calculating natural logarithms.[11] For all complex numbers z satisfying |1 − z| < 1, a simple, though

inefficient, series is

Actually, this is the Taylor series expansion of the natural logarithm at z = 1. It is derived from the geometric series (with |x| < 1)

Integration of both sides yields

and the above expression is obtained by substituting x by 1 − z (so that 1 − x = z).

[edit]More efficient series

A more efficient series is

for z with positive real part.

To derive this series, we begin by substituting −x for x and get

Subtracting, we get

Letting   and thus  , we get

The series converges most quickly if z is close to 1. For high-precision calculations, we can first obtain a low-accuracy

approximation y ≈ ln(z), then let A = z/exp(y), where exp(y) can be calculated using the exponential series, which converges

quickly provided y is not too large. Then ln(z) = y + ln(A), where A is close to 1 as desired. Larger z can be handled by

writing z = a × 10b, whence ln(z) = ln(a) + b × ln(10) (using 10 as an example base). High precision calculations can be first

obtained by low accuracy as mentioned above, this helps in the mathematical process.

[edit]Computation using significands

In most computers, real numbers are modelled by floating points, which are usually stored as

x = m · 2n.

In this representation m is called significand and n is the exponent. Therefore

logb(x) = logb(m) + n logb(2),

so that in order to compute the logarithm of x, it suffices to calculate logb(m) for some m such that 1 ≤ m <  2. Having m in this

range means that the value u = (m − 1)/(m + 1) is always in the range 0 ≤ u < 1/3. Some machines use the significand in the

range 0.5 ≤ m < 1 and in that case the value for u will be in the range −1/3 < u ≤ 0. In either case, the series is even easier to

compute.

The binary (base b = 2) logarithm of a number x can be approximated using the Binary logarithm Algorithm.

Logarithm of a negative or complex number

Main article: Complex logarithm

The principal branch of the complex logarithm, Log(z). The hue of the color shows the argument of Log(z), the saturation (intensity) of the color shows the

absolute value of the complex logarithm.

The above definition of logarithms of positive real numbers can be extended to complex numbers. This generalization known

as complex logarithm requires more care than the logarithm of positive real numbers. Any complex number z can be

represented as z = x + iy, where x and y are real numbers and i is the imaginary unit. The intent of the logarithm is—as with the

natural logarithm of real numbers above— to find a complex number a such that the exponential of a equals z:

ea = z.

Polar form of complex numbers

This can be solved for any z ≠ 0, however there are multiple solutions. To see this, it is convenient to use the polar

form of z, i.e., to express z as

z = r(cos(φ) + i sin(φ))

where   is the absolute value of z and φ = arg(z) an argument of z, that is, is any angle such

thatx = r cos(φ) and y = r sin(φ). Geometrically, the absolute value is the distance of z to the origin and the argument is the

angle between the x-axis and the line passing through the origin and z. The argument φ is not unique: φ' = φ + 2π is an

argument, too, since "winding" around the circle counter-clock-wise once corresponds to adding 2π (360 degrees) to φ.

However, there is exactly one argument φ such that −π < φ and φ ≤ π, called the principal argument and denoted Arg(z) (with a

capital A).

It is a fact proven in complex analysis that z = reiφ. Consequently,

a = ln(r) + i φ

is such that the a-th power of e equals z, so that a qualifies being called logarithm of z. When φ is chosen to be the principal

argument, a is called principal value of the logarithm, denoted Log(z). The principal argument of any positive real number is 0;

hence the principal logarithm of such a number is always real and equals the natural logarithm. The graph at the right depicts

Log(z). The discontinuity, i.e., jump in the hue at the negative part of the x-axis is due to jump of the principal argument at this

locus. This behavior can only be circumvented by dropping the range restriction on φ. Then, however, both the argument

of z and, consequently, its logarithm become multi-valued functions. In fact, any logarithm of z can be shown to be of the form

ln(r) + i (φ + 2nπ),

where φ is the principal argument Arg(z). Analogous formula for principal values of logarithm of products and powers for

complex numbers do in general not hold.

For any complex number b ≠ 0 and 1, the complex logarithm logb(z) with base b is defined as ln(z)/ln(b), the principal value of

which is given by the principal values of ln(z) and ln(b).

[edit]Uses and occurrences

A nautilus displaying a logarithmic spiral.

Logarithms appear in many places, within and outside mathematics. The logarithmic spiral, for example, appears

(approximately) in various guises in nature, such as the shells of nautilus.

[edit]Calculation of products, powers and roots

Logarithms can be used to reduce multiplications and exponentiations to additions. This fact was the historical motivation for

logarithms. The use of logarithms was an essential skill until computers and calculators became available. Indeed the discovery

of logarithms, just before Newton's era, had an impact in the scientific world that can be compared with that of the advent of

computers in the 20th century because it made feasible many calculations that had previously been too laborious.[citation needed]

The product of two numbers c and d can be calculated by the following formula:

c · d = blogb

c · blogb

d = b(logbc + log

bd).

Using a table of logarithms, logbc and logbd can be looked up. After calculating their sum, an easy operation, the antilogarithm

of that sum is looked up in a table, which is the desired product. For manual calculations that demand any appreciable

precision, this process, requiring three lookups and a sum, is much faster than performing the multiplication. To achieve seven

decimal places of accuracy requires a table that fills a single large volume; a table for nine-decimal accuracy occupies a few

shelves. The precision of the approximation can be increased byinterpolating between table entries.

Divisions can be performed similarly:   Moreover,

cd = b(logb

c) · d

reduces the exponetiation to looking up the logarithm of c, multiplying it with d (possibly using the previous method) and looking

up the antilogarithm of the product. Roots   can be calculated this way, too, since  .

One key application of these techniques was celestial navigation. Once the invention of the chronometer made possible the

accurate measurement of longitude at sea, mariners had everything necessary to reduce their navigational computations to

mere additions. A five-digit table of logarithms and a table of the logarithms of trigonometric functions sufficed for most

purposes, and those tables could fit in a small book. Another critical application with even broader impact was the slide rule,

until the 1970s an essential calculating tool for engineers and scientists. The slide rule allows much faster computation than

techniques based on tables, but provides much less precision (although slide rule operations can be chained to calculate

answers to any arbitrary precision).

Logarithmic scaleMain article: Logarithmic scale

Various quantities in science are expressed as logarithms of other quantities, a concept known as logarithmic scale. It applies

in various situations where a given quantity ranges from 1 to 10,000,000, say, in a way such that a change of the value from 1

to 2 is as important as from 100,000 to 200,000. Considering the logarithm instead of the quantity itself reduces such ranges to

smaller ones. Moreover, ratios between different values correspond to differences in their logarithms.

For example, in chemistry, the negative of the base-10 logarithm of the activity of hydronium ions (H3O+, the form H+ takes in

water) is the measure known as pH. The activity of hydronium ions in neutralwater is 10−7 mol/L at 25 °C, hence a pH of

7. Vinegar, on the other hand, 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 10−3 mol/L. In a similar vein, the decibel (symbol dB) is a unit of measure which is the

base-10 logarithm of ratios, such as power levels and voltage levels. It is mostly used in telecommunication, electronics,

and acoustics. In spectrometry and optics, the absorbance unit used to measure optical density is equivalent to −10 dB.

The Richter scale measures earthquake intensity on a base-10 logarithmic scale.

A semi-logarithmic plot of cases and deaths in the 2009 outbreak of influenza A (H1N1).

Semilog graphs depict one, typically the vertical, axis using a logarithmic scaling. This way, exponential functions of the

form f(x) = a · bx appear as a straight line whose slope is proportional to b. At the right, numbers of cases of the swine flu are

shown—the horizontal (time) axis is linear, with the dates evenly spaced, the vertical (cases) axis is logarithmic, with the evenly

spaced divisions being labelled with successive powers of two. In a similar vein, log-log graphs scale both axes logarithmically.

A function f(x) is said to grow logarithmically, if it is (sometimes approximately) proportional to the logarithm. An example is

the harmonic series:

grows about as fast as ln(n).

[edit]Psychology

In psychophysics, the Weber–Fechner law proposes a logarithmic relationship between stimulus and sensation

(though Stevens' power law is typically more accurate). According to the logarithmic responsiveness of the eye to

brightness, the apparent magnitude measures the brightness ofstars logarithmically.

Mathematically untrained individuals tend to estimate numerals with a logarithmic spacing, i.e., the position of a

presented numeral correlates with the logarithm of the given number so that smaller numbers are given more space than

bigger ones. With increasing mathematical training this logarithmic representation becomes more and more linear, a

development that has been found both in Western school children (comparing second to sixth graders)[12] as in

comparison between American and indigene cultures.[13]

[edit]Complexity and entropy

In computer science, logarithms often appear in bounds for computational complexity. For example, to sort N items using

the quicksort algorithm typically requires time proportional to the product N · log(N). Similarly, base-2 logarithms are

used to express the amount of storage space or memory required for a binary representation of a number—with k bits

(each a 0 or a 1) one can represent 2k distinct values, so any natural number N can be represented in no more than

(log2 N) + 1 bits.

Using logarithms, the notion of entropy in information theory is a measure of quantity of information. If a message

recipient may expect any one of N possible messages with equal likelihood, then the amount of information conveyed by

any one such message is quantified as log2 N bits. In the same vein, the concept of entropy also appears

in thermodynamics. Fractal dimension and Hausdorff dimension measure how much space geometric structures occupy.

Example point (straight) line Koch curve Sierpinski triangle plane Apollonian sphere

Hausdorff dimension 0 1 log(4)/log(3) ≈ 1.262 log(3)/log(2) ≈ 1.585 2 ≈ 2.474

[edit]Mathematics

Graph comparing π(x) (red), x / ln x(green) and Li(x) (blue).

Natural logarithms have a tendency to appear in number theory, more specifically in counting primes. For any given

number x, the number of prime numbers less than or equal to x is denoted π(x). In its simples form, the prime number

theorem asserts that π(x) is approximately given by

in the sense that the ratio of π(x) and that fraction approaches 1 when x tends to infinity. This can be rephrased

by saying that the probability that a randomly chosen number between 1 and x is prime is indirectly proportional to

the numbers of decimal digits of x. A far better estimate of π(x) is given by the offset logarithmic integral function

Li(x), defined by

The Riemann hypothesis, one of the oldest open mathematical conjectures, can be stated in terms of comparing π(x) and Li(x).

The Erdős–Kac theorem describing the number of distinct prime factors also involves the natural logarithm.

In geometry the logarithm is used to form the metric for the half-plane model of hyperbolic geometry.

[edit]Statistics

In inferential statistics, the logarithm of the data in a dataset can be used for parametric statistical testing if the original data do

not meet the assumption of normality.

Logarithms appear in Benford's law, a empirical description of the occurrence of digits in certain real-life data sources, such as

heights of buildings. The probability that the first decimal digit of the data in question is d (from 1 to 9) equals

P(d) = log10(d + 1) − log10(d),

irrespective of the unit of measurement. This can be used to detect fraud in accounting.[14]

[edit]Music

Logarithms appear in measuring musical intervals: the interval between two notes in semitones is the base-21/12 logarithm of

the frequency ratio. For finer encoding, for example for non-equal temperaments, intervals are also expressed

in cents (hundredths of an equally-tempered semitone). The interval between two notes in cents is the base-21/1200 logarithm of the frequency ratio (or 1200 times the base-2 logarithm). The table below lists some musical intervals together with the frequency ratios and their logarithms.

Interval (two tones are played at the

same time)

1/72 tone play (help·info)

Semitone play

Just major third

play

Major third play Tritone play

Octave

play

Frequency ratio r 2

, i.e., corresponding number of semitones

1/6 1 ≈ 3.86 4 6 12

, i.e., corresponding number of cents

16.67 100≈

386.31

400 6001200

[edit]Related operations and generalizations

The cologarithm of a number is the logarithm of the reciprocal of the number: cologb(x) = logb(1/x) = −logb(x). This terminology is

found primarily in older books.[15]

The antilogarithm function antilogb(y) is the inverse function of the logarithm function logb(x); it can be written in closed form

as by. The antilog notation was common before the advent of modern calculators and computers: tables of antilogarithms to the

base 10 were useful in carrying out computations by hand.[16] Today's applications of antilogarithms include certain electronic

circuit components known asantilog amplifiers,[17] or the calculation of equilibrium constants of reactions involving electrode

potentials.

The double or iterated logarithm, ln(ln(x)), is the inverse function of the double exponential function. The super- or hyper-4-

logarithm is the inverse function of tetration. The super-logarithm of x grows even more slowly than the double logarithm for

large x.

The Lambert W function is the inverse function of ƒ(w) = wew. Polylogarithm is a generalization of the logarithm defined by

For s = 1, it is related to the logarithm via Li1(z) = −ln(1 − z). Forz = 1, on the other hand, Lis(1) yields the Riemann zeta

function ζ(s).

From the pure mathematical perspective, the identity log(cd) = log(c) + log(d) expresses an isomorphism between

the multiplicative group of the positive real numbers and the group of all the reals under addition. Logarithmic functions are the

only continuous isomorphisms from the multiplicative group of positive real numbers to the additive group of real numbers. The

logarithm function can be extended to aHaar measure in the topological group of positive real numbers under multiplication.

The discrete logarithm is a related notion in the theory of finite groups. It involves solving the equation bn = x, where b and x are

elements of the group, and n is an integer specifying a power in the group operation. For some finite groups, it is believed that

the discrete logarithm is very hard to calculate, whereas discrete exponentials are quite easy. This asymmetry has applications

in public key cryptography.[18]

Logarithms can be defined for p -adic numbers ,[19] quaternions and octonions. The logarithm of a matrix is the inverse of

the matrix exponential.

[

A more modern definition and explanation from 1866 A Dictionary of Science, Literature, & Art: Comprising the

Definitions and Derivations of the Scientific Terms in General Use, together with the History and Descriptions of the

Scientific Principles of Nearly Every Branch of Human Knowledge

Michael Stifel published Arithmetica integra in Nuremberg in 1544 which contains his

discovery of logarithms.

The method of logarithms was publicly propounded in 1614, in a book entitled Mirifici

Logarithmorum Canonis Descriptio, by John Napier, Baron of Merchiston, in Scotland,

[20] (Joost Bürgi independently discovered logarithms; however, he did not publish his

discovery until four years after Napier). Early resistance to the use of logarithms was muted

by Kepler's enthusiastic support and his publication of a clear and impeccable explanation of

how they worked.[21]

Their use contributed to the advance of science, and especially of astronomy, by making

some difficult calculations possible. Prior to the advent of calculators and computers, they

were used constantly in surveying, navigation, and other branches of practical mathematics.

It supplanted the more involved method of prosthaphaeresis, which relied on trigonometric

identities as a quick method of computing products. Besides the utility of the logarithm

concept in computation, the natural logarithm presented a solution to the problem

of quadrature of a hyperbolic sector at the hand ofGregoire de Saint-Vincent in 1647.

At first, Napier called logarithms "artificial numbers" and antilogarithms "natural numbers".

Later, Napier formed the word logarithm to mean a number that indicates a

ratio: λόγος (logos) meaning proportion, and ἀριθμός (arithmos) meaning number. Napier

chose that because the difference of two logarithms determines the ratio of the numbers they

represent, so that an arithmetic series of logarithms corresponds to a geometric series of

numbers. The term antilogarithm was introduced in the late 17th century and, while never

used extensively in mathematics, persisted in collections of tables until they fell into disuse.

Napier did not use a base as we now understand it, but his logarithms were, up to a scaling

factor, effectively to base 1/e. For interpolation purposes and ease of calculation, it is useful

to make the ratio r in the geometric series close to 1. Napier chose r = 1 − 10−7 = 0.999999

(Bürgi choser = 1 + 10−4 = 1.0001). Napier's original logarithms did not have log 1 = 0 but

rather log 107 = 0. Thus if N is a number and L is its logarithm as calculated by

Napier, N = 107(1 − 10−7) L. Since (1 − 10−7)107 is approximately 1/e, this

makes L / 107 approximately equal to log1/e N/107.[9]

Part of a 20th-century table of common logarithms in the reference bookAbramowitz and Stegun.

Page of a 1912 table, which was common in schools.

[edit]Tables of logarithmsMain article: Table of logarithms

Prior to the advent of computers and calculators, using logarithms for practical purposes was

done with tables of logarithms, which contain both logarithms and antilogarithms with respect

to some fixed base.

When the purpose was to facilitate arithmetic computations, base 10 was used, and

logarithms of numbers between 1 and 10 by small increments (e.g. 0.01 or 0.001) appeared.

There was no need for logarithms of other numbers (above 10 or below 1) because moving

the decimal point corresponds to simply adding an integer to the logarithm. See Common

logarithm for further details.

The following table lists the major tables of logarithms compiled in history: