Logarithm

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 log_b(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.

Logarithm of positive real numbers

Definition

The logarithm of a number y with respect to a number b is the power to which b has to be raised in order to give y. In symbols, the logarithm is the number x satisfying the following equation:

b^x = y.

The logarithm x is denoted log_b(y). (Some European countries write ^blog(y) instead.^[1]) The number b is called base. For b = 2, for example, this means

log₂(8) = 3,

since 2³ = 2 · 2 · 2 = 8. Another example is

since

For the logarithm to exist, one has to require the base b to be a positive real number unequal to 1. Secondly, y has to be a positive real number. These restrictions are necessary because of the following: 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 one solution. Secondly, to avoid ambiguity, it is necessary to ensure that there is only one solution. Both points are satisfied with the said restrictions on b and y, but are not without these restrictions. This is explained below.

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

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

Logarithmic identities

There are three important formulas, sometimes called logarithmic identities, relating various logarithms to one another. The first is about the logarithm of a product, the second about logarithms of powers and the third involves logarithms with respect to different bases.

Logarithm of a product

The logarithm of a product is the sum of the two logarithms. That is, for any two positive real numbers x and y, and a given positive base b, the following identity holds:

log_b(x · y) = log_b(x) + log_b(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:

b^s · b^t = b^{s + t.}

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

log_b(b^s · b^t) = log_b(b^{s + t}) = s + t = log_b(b^s) + log_b(b^t).

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 a power

The logarithm of the p-th power of a number x is p times the logarithm of that number. In symbols:

log_b(x^p) = p log_b(x).

As an example,

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

This formula can be proven as follows: the logarithm of x is the number to which the base b has to be raised in order to yield x. In other words, the following identity holds:

x = b^log_b(x).

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

x^p = (b^log_b(x))^p = b^{p · log_b(x)}.

(At this point, the identity (d^e)^f = d^{e · f} was used, where d, e and f are positive real numbers.) Thus, the log_b of the left hand side, log_b(x^p), and of the right hand side, p · log_b(x), agree. The sought formula is proven.

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 third important rule for calculating logarithms is the following formula calculating the logarithm of a fixed number x to one base in terms of the one to another base:

This is shown as follows: the left hand side of the above is the unique number a such that b^a = x. Therefore

log_k(x) = log_k(b^a) = a · log_k(b).

The general restriction b ≠ 1 implies log_kb ≠ 0, since b⁰ = 1. Thus, dividing the preceding equation by log_kb 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 log_k x.

Particular 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 example, it has been argued that common logarithms are easier to deal with than other ones, since in the decimal number system the powers of 10 have a simple representation.^[2] Mathematical analysis, on the other hand, prefers the base b = e because of the "natural" properties explained below.

The following table lists common notations for logarithms to these bases and the fields where they are used. Instead of writing log_b(x), it is common in many disciplines to omit the base, log(x), when intended base can be determined from context. The notations suggested by the International Organization for Standardization (ISO 31-11) are underlined in the table below.^[3] Given a number n and its logarithm log_b(n), the base b can be determined by the following formula:

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

Analytic properties

Domain of definition

In the above definition, it was required that b be positive, but unequal to 1, and that y be positive. If these assumptions are met, the equation

b^x = y

has exactly one solution x. To prove this, it is helpful to consider the cases b > 1 and 0 < b < 1 separately. If b > 1, b^x is a continuous, strictly monotonic function from the reals to the positive reals, so it has an inverse function from the positive reals to the reals, which is denoted by log_b (x). If 0 < b < 1, then the identity

can be used to define

Logarithm as a function

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

In the more highbrow language of calculus, the above proof amounts to saying that the function f(x) = b^x is strictly increasing, takes arbitrarily big and small positive values and 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 of elementary calculus ensure that there is indeed exactly one solution x to the equation

f(x) = b^x = y,

for any given positive y.

The expression log_b(x) depends on both b and x, but the term logarithm function (or logarithmic function) refers to a function of the form log_b(x) in which the base b is fixed and x is variable, thus yielding a function that assigns to any x its logarithm log_b(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 b^x: a point (t, u = b^(t)) on the graph of the exponential function yields a point (u, t = log_bu) 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 log_b(x) has no sharp "corners").

Integral representation of the natural logarithm

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

When b = e (Euler's number), the natural logarithm ln(t) = log_e(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 = 1 to x = t. This is depicted at the right. Some authors use the right hand side of this equation as a definition of the natural logarithm.^[8] 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(t^r) = r ln(t) is derived similarly:

The second equality is using a change of variables, w := x^1/r, while the third equality follows from integration by substitution.

Not only the integral over 1/x is tied to the logarithm, the harmonic series,

also is: as n tends to infinity, the series

converges to a number known as Euler–Mascheroni constant. Little is known about it—not even whether it is a rational number or not.^[9]

Derivative and antiderivative

The derivative of the natural logarithm is given by

This can be derived either from the definition as the inverse function of exp(x), using the chain rule,^[10] or from the above integral formula, using the fundamental theorem of calculus. More generally 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

Derivatives and antiderivatives of logarithms to other bases can be derived therefrom using the formula for change of bases.

Calculation

Taylor series

The Taylor series of ln(z) at z = 1. The animation shows the first 10 approximations.

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 (which is the Taylor series of 1/(1 − x)), which converges for |x| < 1:

Taking the indefinite integral of both sides yields

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

More efficient series

Another series is

for complex numbers z with positive real part. This series can be derived from the above Taylor series. The series converges most quickly if z is close to 1. For example, for z = 1.5, the first three terms of this series approximate ln(1.5) with an error of about 3·10⁻⁶. (For comparison, the above Taylor series needs 13 terms to achieve that precision.) The quick convergence for z close to 1 can be taken advantage of in the following way: given a low-accuracy approximation y ≈ ln(z) and putting A = z/exp(y), the logarithm of z is

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

The better the initial approximation y is, the closer A is to 1, so its logarithm can be calculated efficiently. The calculation of A can be done using the exponential series, which converges quickly provided y is not too large. Calculating the logarithm of larger z can be reduced to smaller values of z by writing z = a · 10^b, so that ln(z) = ln(a) + b · ln(10).

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

log_b(x) = log_b(m) + n log_b(2),

so that in order to compute the logarithm of x, it suffices to calculate log_b(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

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:

e^a = 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 that x = r cos(φ) and y = r sin(φ).^[12] 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.^[13] (The normalization 0 ≤ φ < 2π for the principal argument also appears in the literature.^[14])

It is a fact proven in complex analysis that z = re^iφ. 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 log_b(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).

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.^[15] Logarithms also appear in formulas in various sciences, such as the Tsiolkovsky rocket equation or Fenske equation.

The logarithm of a positive integer x to a given base shows how many digits are needed to write that integer in that base. For instance, the common logarithm log₁₀(x) is linked to the number n of numerical digits of x: n is the smallest integer strictly bigger than log₁₀(x), i.e., the one satisfying the inequalities:

n − 1 ≤ log₁₀(x) < n.

Similarly, the number of binary digits (or bits) needed to store a positive integer x is the smallest integer strictly bigger than log₂(x).

Calculation of products, powers and roots

Logarithms can be used to reduce multiplications and exponentiations to additions, thus making feasible many calculations that are otherwise too laborious. This fact was the historical motivation for logarithms. The use of logarithms was an essential skill until computers and calculators became available.

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

c · d = b^log_bc · b^log_bd = b^{(log_bc + log_bd)}.

Using a table of logarithms, log_bc and log_bd 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 by interpolating between table entries.

Divisions can be performed similarly: Moreover,

c^d = b^{(log_bc) · d}

reduces the exponentiation 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).

The C and D scales on this slide rule are marked off at positions corresponding to the logarithms of the numbers shown. By mechanically adding the logarithms of 1.3 and 2, the cursor shows the product is 2.6.

Logarithmic scale

A logarithmic chart depicting the value of one Goldmark in Reichsmark during the German hyperinflation in the 1920s.

Various quantities in science are expressed as logarithms of other quantities, a concept known as logarithmic scale. For example, the Richter scale measures the strength of an earthquake by taking the common logarithm of the energy emitted at the earthquake. Thus, an increase in energy by a factor of 10 adds one to the Richter magnitude; a 100-fold energy results in +2 in the magnitude etc.^[16] Considering the logarithm instead of the value, here the energy of the quake, reduces the range of possible values to a much smaller range. A second example is the pH in chemistry: it is the negative of the base-10 logarithm of the activity of hydronium ions (H₃O⁺, the form H⁺ takes in water).^[17] The activity of hydronium ions in neutral water 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. For example it is used to quantify the loss of voltage levels in transmitting electrical signals^[18] or to describe power levels of sounds in acoustics.^[19] In spectrometry and optics, the absorbance unit used to measure optical density is equivalent to −10 dB. The apparent magnitude measures the brightness of stars logarithmically.

Semilog graphs or log-linear graphs use this concept for visualization: one axis, typically the vertical one, is scaled logarithmically. This way, exponential functions of the form f(x) = a · b^x appear as a straight line whose slope is proportional to b. In a similar vein, log-log graphs scale both axes logarithmically.^[20]

Psychology

In psychophysics, the Weber–Fechner law proposes a logarithmic relationship between stimulus and sensation (though Stevens' power law is typically more accurate). The logarithm appears in situations when the smallest change ΔS of some stimulus S that an observer can still notice is proportional to S, by the above relation of the natural logarithm and the integral over dS / S. (Similar formulas also appear in other sciences, such as the Clausius–Clapeyron relation.) Hick's law proposes a logarithmic relation between the time individuals take for choosing an alternative and the number of choices they have.^[21]

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, as confirmed both in Western school children, comparing second to sixth graders,^[22] as well as in comparison between American and indigene cultures.^[23]

Complexity and entropy

A function f(x) is said to grow logarithmically, if it is (sometimes approximately) proportional to the logarithm. (Biology, in describing growth of organisms, uses this term for an exponential function, though.^[24]) It is irrelevant to which base the logarithm is taken, since choosing a different base amounts to multiplying the result by a constant, as follows from the formula above. For example, any natural number N can be represented in binary form in no more than (log₂(N)) + 1 bits. I.e., the amount of hard disk space on a computer grows logarithmically as a function of the size of the number to store. Corresponding calculations carried out using log_e will lead to results in nats which may lack this intuitive interpretation. However, the change amounts to a factor of log_e2≈0.69—twice as many values can be encoded with one additional bit, which corresponds to an increase of about 0.69 nats. A similar example is the relation of decibel (dB), using a common logarithm (b = 10) vis-à-vis neper, based on a natural logarithm.

Some disciplines disregard the factor resulting from choosing different bases of logarithms. For example, complexity theory, a branch of computer science, describes the asymptotic behavior of algorithms by making statements like "the behavior of the algorithm is logarithmic". For example, sorting N items using the quicksort algorithm typically requires time proportional to the product N · log(N). Using big O notation, the complexity is denoted as

O(N · log(N)),

which is a lot slower a growth than, say the complexity O(N²) of bubble sort.

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.^[25] 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

Pure mathematics

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

Natural logarithms have a tendency to appear in number theory. For any given number x, the number of prime numbers less than or equal to x is denoted π(x). In its simplest 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.^[26] 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).^[27] The Erdős–Kac theorem describing the number of distinct prime factors also involves the natural logarithm.

By the formula calculating logarithms of products, the logarithm of n factorial, n! = 1 · 2 · ... · n, is given by

ln(n!) = ln(1) + ln(2) + ... + ln(n).

This can be used to obtain Stirling's formula, an approximation for n! for large n.^[28]

In geometry the logarithm is used to calculate the distance on the Poincaré half-plane model of the hyperbolic geometry.

Statistics

Distribution of first digits (in %, red bars) in the population of the 237 countries of the world. Black dots indicate the distribution predicted by Benford's law.

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, an 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

log₁₀(d + 1) − log₁₀(d),

irrespective of the unit of measurement.^[29] Thus according to that law, about 30% of the data can be expected to have 1 as first digit, 18% start with 2 etc. Deviations from this pattern can be used to detect fraud in accounting.^[30]

Music

Logarithms appear in the encoding of musical tones. In equal temperament, the frequency ratio depends only on the interval between two tones, not on the specific frequency, or pitch of the individual tones. Therefore, logarithms can be used to describe the intervals: the interval between two notes in semitones is the base-2^1/12 logarithm of the frequency ratio.^[31] For finer encoding, as it is needed 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-2^1/1200 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	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	600	1200

Related operations and generalizations

The cologarithm of a number is the logarithm of the reciprocal of the number: colog_b(x) = log_b(1/x) = −log_b(x). This terminology is found primarily in older books.^[32] The antilogarithm function antilog_b(y) is the inverse function of the logarithm function log_b(x); it can be written in closed form as b^y.^[33] 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.^[34] Today's applications of antilogarithms include certain electronic circuit components known as antilog amplifiers,^[35] 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) = we^w. Polylogarithm is a generalization of the logarithm defined by

For s = 1, it is related to the logarithm via Li₁(z) = −ln(1 − z). For z = 1, on the other hand, Li_s(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.^[36] By means of that isomorphism, the Lebesgue measure dx on R corresponds to the Haar measure dx/x on the positive reals.^[37] In complex analysis and algebraic geometry, differential forms of the form (d log(f) =) df/f are known as forms with logarithmic poles.^[38]

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.^[39]

Logarithms can be defined for p-adic numbers,^[40] quaternions and octonions. The logarithm of a matrix is the inverse of the matrix exponential.

History

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

The Indian mathematician Virasena worked with the concept of ardhaccheda: the number of times a number could be halved; effectively similar to the integer part of logarithms to base 2. He described various relations using this operation as well as working with logarithms in base 3 (trakacheda) and base 4 (caturthacheda).^[41][42]

Michael Stifel published Arithmetica integra in Nuremberg in 1544; it contains a table of integers and powers of 2 that some have considered to be an early version of a logarithmic table.^[43][44]

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,^[45] (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.^[46]

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 of Gregoire 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 chose r = 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 log_1/e N/10⁷.^[4]

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

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

Tables 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:

Briggs and Vlacq also published original tables of the logarithms of the trigonometric functions.

See also

Notes

^ a: 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.^[49] In fact, the notation was invented by a mathematician Irving Stringham.^[50][51]

References

• Bateman, P. T.; Diamond, Harold G. (2004), Analytic number theory: an introductory course, World Scientific, ISBN 9789812560803, OCLC 492669517 
• Shirali, Shailesh (2002), A Primer on Logarithms, Universities Press, ISBN 9788173714146 

External links

• Logarithms on geoset.info
• Jost Burgi, Swiss Inventor of Logarithms
• Translation of Napier's work on logarithms
• Algorithm for determining Log values for any base