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).

In mathematics, the logarithm of a number to a given base is the 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 10 to the power of 3 is 1000: 10³ = 1000. The logarithm of x to the base b is written log_b(x) such as log₁₀(1000) = 3.

By the following formulas, logarithms reduce multiplication to addition and exponentiation to products:

logb(x · y)	=	logb(x) + logb(y),
logb(xp)	=	p logb(x).

Three particular values for the base b are most common. The natural logarithm, the one with base b = e, occurs in calculus since its derivative is 1/x. The logarithm to base b = 10 is called common logarithm, while the base b = 2 gives rise to the binary logarithm.

The invention of logarithms is due to John Napier in the early 17th century. Before calculators became available, via logarithm tables, logarithms were crucial to simplifying scientific calculations. Today's applications of logarithms are numerous. Logarithmic scale reduces wide-ranging quantities to smaller scopes; this is applied in the Richter scale, for example. In addition to being a standard function used in various scientific formulas, logarithms appear in determining the complexity of algorithms and of fractals. They also form the mathematical backbone of musical intervals and some models in psychophysics, and have been used in forensic accounting. Logarithms are also closely related to questions revolving around counting prime numbers.

Different routines are available to calculate logarithms, some designed for speedy calculations, other ones for high accuracy. Logarithms have been generalized in various ways. The complex logarithm applies to complex numbers instead of real numbers. The discrete logarithm is an important primer in public-key cryptography.

Logarithm of positive real numbers

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. The number b is called base. In symbols, the logarithm is the number x solving the following equation:

b^x = y.

The logarithm x is denoted log_b(y). For b = 2, for example, this means

log₂(8) = 3,

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

$\log _{2}\left({\frac {1}{2}}\right)=-1,\,$

since

$2^{-1}={\frac {1}{2^{1}}}={\frac {1}{2}}.$

The first equality is because a⁻¹ is the reciprocal of a, 1 / a, for any number a unequal to zero. A third example: log₁₀(150) is approximately 2.176. Indeed, 10² = 100 and 10³ = 1000. As 150 lies between 100 and 1000, its logarithm lies between 2 and 3. Ways of calculating the logarithm are explained further down.

The logarithm log_b(y) is defined for any positive number y and any positive base b which is unequal to 1. These restrictions are explained below.

Logarithmic identities

Main article: 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 base b, the following equality 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. For y = b, it yields

log_b(x · b) = log_b(x) + 1,

since log_b(b) is 1, because b¹ = b. In other words, multiplying the number x by the base b amounts to adding one to its logarithm. In general, that identity is proven using the following relation of powers and multiplication:

b^s · b^t = b^{s + t}.

Substituting the particular values s = log_b(x) and t = log_b(y) gives

b^log_b(x) · b^log_b(y) = b^{log_b(x) + log_b(y)}.

The left hand side agrees with x · y. Indeed, the logarithm of x is the number to which the base b has to be raised in order to yield x. That is to say:

x = b^log_b(x)

and a similar equality holds with y in place of x. Taking the base-b-logarithm of both sides yields

log_b(x · y) = log_b(b^{log_b(x) + log_b(y)}) = log_b(x) + log_b(y).

Logarithms also convert divisions to subtractions:

$\log _{b}\left({\frac {x}{y}}\right)=\log _{b}(x)-\log _{b}(y),$

for example

$\log _{2}(16)=\log _{2}\left({\frac {64}{4}}\right)=\log _{2}(64)-\log _{2}(4)=6-2=4.$

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).

Also, logarithms relate roots to division:

$\log _{b}({\sqrt[{p}]{x}})={\frac {\log _{b}(x)}{p}}.\,$

As an example, the logarithm to base b = 2 of 64 is 6, since 2⁶ = 64. But 64 is also 4³ and indeed 3 · log₂(4) = 3 · 2 equals 6. This formula can be derived by raising the afore-mentioned identity

x = b^log_b(x)

to the p-th power (exponentiation), which gives

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

For the second equality, the identity (d^e)^f = d^{e · f} was used. 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.

Change of base

The following formula relates the logarithm of a fixed number x to one base in terms of the one to another base:

$\log _{b}{x}={\frac {\log _{k}{x}}{\log _{k}{b}}}.\,$

This is actually a consequence of the previous rule, as the following proof shows: taking the base-k-logarithm of the above-mentioned identity,

x = b^log_b(x),

yields

log_k(x) = log_k(b^log_b(x)).

By the previous rule, the right hand term equals log_b(x) · log_k(b). Dividing the preceding equation by log_k(b) shows the change-of-base formula. The division is possible since log_kb ≠ 0. Indeed, this logarithm cannot be zero, since k⁰ = 1, but b was supposed to be different than 1.

As a practical consequence, logarithms with respect to any base k can be calculated with a calculator, if logarithms to any base b (often b = 10 or b = e) are available: log_k(x) = log₁₀(x) / log₁₀(k).

Particular bases

While the definition of logarithm applies to any positive real number b (excluding 1), 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, base-10 logarithms are easier than other bases to use with a table of logarithms or a slide rule for computations when electronic means are not available, since in the decimal number system the powers of 10 have a simple representation. 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 and to write log(x), when the intended base can be determined from the context. The notations suggested by the International Organization for Standardization (ISO 31-11) are underlined in the table below. Given a number n and its logarithm log_b(n), the base b can be determined by the following formula:

$b=n^{\frac {1}{\log _{b}(n)}}.$

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

Base b	Name for logb(x)	Notations for logb(x)	Used in
2	binary logarithm	lb(x), ld(x), log(x) (in computer science), lg(x)	computer science, information theory
e	natural logarithm	ln(x), log(x) (in mathematics and many programming languages including C, Java, Haskell, and BASIC</ref>)	mathematical analysis, physics, chemistry, statistics, economics and some engineering fields
10	common logarithm	lg(x), log(x) (in engineering, biology, astronomy),	various engineering fields, (see decibel and see below), logarithm tables, handheld calculators

Analytic properties

Logarithm as a function

The following discussion of logarithms uses the notion of function. In a nutshell, a function is a datum that assigns to a given number another number. The first one is called variable to emphasize the idea that it can take different values.

The expression log_b(x) depends on both the base b and on x. The base b is usually regarded as fixed. Therefore the logarithm only depends on the variable x, a positive real number. Assigning to x its logarithm log_b(x) therefore is a function. It is called logarithm function or logarithmic function or even just logarithm.

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).

The above definition of logarithms was done indirectly by means of the exponential function f(x) = b^x: the logarithm (to base b) of y is the solution x to the equation

f(x) = y.

A rigorous proof that this equation actually has a solution and that this solution is unique requires some elementary calculus, specifically the intermediate value theorem. This theorem says that a continuous function that takes two values m and n also takes any intermediate value, that is, any value that lies between m and n. A function is continuous if it does not "jump", that is, if its graph can be drawn without lifting the pen. This property can be shown to hold for the exponential function f(x). Moreover, it takes arbitrarily big and arbitrarily small positive values, so that any number y can be boxed by y₀ and y₁ which are values of the function f. Thus, the intermediate value theorem ensures that the equation

f(x) = y

has a solution. Moreover, as the function f is strictly increasing (for b > 1), or strictly decreasing (for 0 < b < 1), there is only one solution to this equation.

A compact way of rephrasing the point that the base-b logarithm of y is the solution x to the equation f(x) = b^x = y is to say that the logarithm function is the inverse function of the exponential function. Inverse functions are closely related to the original functions: the graphs of the two correspond to each other upon reflecting them at the diagonal line x = y, as shown at the right: 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. Moreover, analytic properties of the function pass to its inverse function. Thus, as the exponential function f(x) = b^x is continuous and differentiable, so is its inverse function, log_b(x). Roughly speaking, a differentiable function is one whose graph has no sharp "corners".

Derivative and antiderivative

Further information: List of integrals of logarithmic functions

The area of the hyperbolic sector equals ln(b) − ln(a).

The derivative of the natural logarithm ln(x) = log_e(x) is given by

${\frac {d}{dx}}\ln(x)={\frac {1}{x}}.$

This can be derived from the definition as the inverse function of e^x, using the chain rule. This implies that the antiderivative of 1/x is ln(x) + C. An early application of this fact was the quadrature of a hyperbolic sector, as shown at the right, by de Saint-Vincent in 1647. The derivative with a generalised functional argument f(x) is

${\frac {d}{dx}}\ln(f(x))={\frac {f'(x)}{f(x)}}.$

For this reason the quotient at right hand side is called logarithmic derivative of f. The antiderivative of the natural logarithm ln(x) is

$\int \ln(x)\,dx=x\ln(x)-x+C.$

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

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).

The natural logarithm satisfies the following identity:

$\ln(t)=\int _{1}^{t}{\frac {1}{x}}\,dx.$

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. The formula is a consequence of the fundamental theorem of calculus and the above formula for the derivative of ln(x). Some authors actually use the right hand side of this equation as a definition of the natural logarithm and derive the formulas concerning logarithms of products and powers mentioned above from this definition. The product formula ln(tu) = ln(t) + ln(u) is deduced in the following way:

$\ln(tu)=\int _{1}^{tu}{\frac {1}{x}}\,dx\ {\stackrel {(1)}{=}}\int _{1}^{t}{\frac {1}{x}}\,dx+\int _{t}^{tu}{\frac {1}{x}}\,dx=\ln(t)+\int _{t}^{tu}{\frac {1}{x}}\,dx\ {\stackrel {(2)}{=}}\ln(t)+\int _{1}^{u}{\frac {1}{w}}\,dw=\ln(t)+\ln(u).$

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:

$\ln(t^{r})=\int _{1}^{t^{r}}{\frac {dx}{x}}=\int _{1}^{t}{\frac {1}{w^{r}}}\left(rw^{r-1}\,dw\right)=r\int _{1}^{t}{\frac {1}{w}}\,dw=r\ln(t).$

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

The sum over the reciprocals of natural numbers, the so-called harmonic series

$1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}=\sum _{k=1}^{n}{\frac {1}{k}}.$

is also closely tied to the natural logarithm: as n tends to infinity, the difference

$\sum _{k=1}^{n}{\frac {1}{k}}-\ln(n)$

converges (i.e., gets arbitrarily close) to a number known as Euler–Mascheroni constant. Little is known about it—not even whether it is a rational number or not. This relation is used in the performance analysis of algorithms such as quicksort, see below.

Calculation

There are a variety of ways to calculate logarithms for practical purposes. One method uses power series, that is to say a sequence of polynomials whose values get arbitrarily close to the exact value of the logarithm. For high precision calculation of the natural logarithm, an approximation formula based on the arithmetic-geometric mean is used. Algorithms involving lookups of precalculated tables are used when emphasis is on speed rather than high accuracy. Moreover, the binary logarithm algorithm calculates lb(x) recursively based on repeated squarings of x, taking advantage of the relation

log₂(x²) = 2 log₂(x).

Finally, based on quick ways to calculate exponentials e^y, the natural logarithm of x, that is the solution a to

e^a − x = 0

can also efficiently be calculated using Newton's method.

While in some cases—such as log₁₀(10.000) = 4—logarithms can be easy to compute, they generally take less simple values: by the Gelfond–Schneider theorem, given two algebraic numbers a and b such as ${\sqrt[{3}]{2}}$ or ${\sqrt {-5+{\sqrt {3/13}}}}$, the ratio γ = ln(a) / ln(b) is either a rational number p / q (in which case a^q = b^p) or transcendental. The latter means that for all n and all rational numbers r₀, ..., r_n the expression

r_n γⁿ + r_{n − 1} γ^{n − 1} + ... + r₁ γ + r₀

is nonzero. Related questions in transcendence theory such as linear forms in logarithms are a matter of current research.

Taylor series

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

For real numbers z satisfying 0 < z < 2, the natural logarithm of z can be written as

${\begin{aligned}\ln(z)&=(z-1)-{\frac {(z-1)^{2}}{2}}+{\frac {(z-1)^{3}}{3}}-{\frac {(z-1)^{4}}{4}}+\cdots \\&=\sum _{n=1}^{\infty }(-1)^{n+1}{\frac {(z-1)^{n}}{n}}=-\sum _{n=1}^{\infty }{\frac {(1-z)^{n}}{n}}.\end{aligned}}$

The infinite sum means that the logarithm of z is approximated by the sums

$-(1-z)-{\frac {(1-z)^{2}}{2}}-\cdots -{\frac {(1-z)^{n}}{n}}$

to arbitrary precision, provided n big is enough. In the parlance of elementary calculus, ln(z) is the limit of the sequence of these sums. This series is the Taylor series expansion of the natural logarithm at z = 1.

More efficient series

Another series is

$\ln(z)=2\left({\frac {z-1}{z+1}}+{\frac {1}{3}}{\left({\frac {z-1}{z+1}}\right)}^{3}+{\frac {1}{5}}{\left({\frac {z-1}{z+1}}\right)}^{5}+\cdots \right)=2\sum _{n=0}^{\infty }{\frac {1}{2n+1}}{\left({\frac {z-1}{z+1}}\right)}^{2n+1}$

for complex numbers z with positive real part. This series can be derived from the above Taylor series. It converges more quickly than the Taylor series, especially if z is close to 1. For example, for z = 1.5, the first three terms of the second series approximate ln(1.5) with an error of about 3·10⁻⁶. 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).

Arithmetic-geometric mean approximation

The natural logarithm ln(x) is approximated to a precision of 2^−p (or p precise bits) by

${\frac {\pi }{2M(1,4/s)}}-m\ln 2.$

Here M denotes the arithmetic-geometric mean and m is chosen so that s = x / 2^m is bigger than 2^p/2. The constants π and ln(2) can be calculated with particular series. The formula goes back to Gauss.

Complex logarithm

Main article: Complex logarithm

Polar form of complex numbers

The complex logarithm is a generalization of the above definition of logarithms of positive real numbers to complex numbers. Complex numbers are commonly represented as z = x + iy, where x and y are real numbers and i is the imaginary unit. As was discussed above, given any real number z ≠ 1, the equation

e^a = z

has exactly one real solution a. However, there are infinitely many complex numbers a solving this equation, i.e., multiple complex numbers a whose exponential equals z. This causes complex logarithms to be different from real ones.

The solutions of the above equation are most readily described using the polar form of z. It encodes a complex number z by its absolute value, that is, the distance r to the origin, and the argument φ, the angle between the line connecting the origin and z and the x-axis. In terms of the trigonometric functions sine and cosine, r and φ are such that

z = r · cos(φ) + i · r · sin(φ).

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 absolute value r is uniquely determined by z by the formula $r=|z|={\sqrt {x^{2}+y^{2}}}$ but there are multiple numbers φ such that the preceding equation holds—given one such φ,

φ' = φ + 2π

also satisfies the preceding equation. Adding 2π or 360 degrees to the argument corresponds to "winding" around the circle counter-clock-wise by an angle of 2π. However, there is exactly one argument φ satisfying −π < φ and φ ≤ π. It is called the principal argument and denoted Arg(z), with a capital A. (The normalization 0 ≤ Arg(z) < 2π also appears in the literature.) It is a fact proven in complex analysis that

z = re^iφ.

Consequently, if φ is the principal argument Arg(z), the number number

a = ln(r) + i · (φ + 2nπ)

is such that the a-th power of e equals z, for any integer n. Accordingly, a is called complex logarithm of z. If n = 0, 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 a real number and equals the natural logarithm as defined above. In contrast to the real case, analogous formula for principal values of logarithm of products and powers for complex numbers do in general not hold.

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 the jump of the principal argument at this locus. This behavior can only be circumvented by dropping the range restriction on φ. Then the argument of z and, consequently, its logarithm become multi-valued functions.

Uses and occurrences

A nautilus displaying a logarithmic spiral.

Logarithms have many applications, both within and outside mathematics. The logarithmic spiral, for example, appears (approximately) in various guises in nature, such as the shells of nautilus. Logarithms also occur in various scientific formulas, such as the Tsiolkovsky rocket equation, the Fenske equation, or the Nernst 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).

A tiling of the Poincaré disk of hyperbolic geometry by ideal triangles—their edges are shortest "lines" in hyperbolic geometry.

In hyperbolic geometry the logarithm is used in measuring the distances between points. In the Poincaré disk model, this causes the shortest connections between any two points to be either segments of circles perpendicular to the boundary of the disk or lines through the origin, the center of the circle.

Logarithmic scale

Main article: Logarithmic scale

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

Various scientific quantities 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. This way, large-scaled quantities are reduced to much smaller ranges. 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). The activity of hydronium ions in neutral water is 10⁻⁷ mol·L⁻¹ at 25 °C, hence a pH of 7. Vinegar (typically around 5% w/v aqueous solution of acetic acid), 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⁻¹. 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 or to describe power levels of sounds in acoustics. 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 straight lines with slope proportional to b. In a similar vein, log-log graphs scale both axes logarithmically.

Psychology

In psychophysics, the Weber–Fechner law proposes a logarithmic relationship between stimulus and sensation. (However, it is often considered to be less accurate than Stevens' power law.) According to that model, the smallest change ΔS of some stimulus S that an observer can still notice is proportional to S. This gives rise to a logarithmic relation by the above relation of the natural logarithm and the integral over dS / S. Hick's law proposes a logarithmic relation between the time individuals take for choosing an alternative and the number of choices they have.

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 gradually superseeded by a linear one—a finding confirmed both in comparing second to sixth grade Western school children, as well as in comparison between American and indigene cultures.

Probability theory and 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.

Logarithms also arise in probability theory: tossing a coin repeatedly, it is known (by the law of large numbers), that the "heads"-vs.-"tails"-ratio is approaching 0.5 as the number of tosses increases. The fluctuations of this ratio around the limiting value are quantified by the law of the iterated logarithm.

Logarithms are used in the process of maximum likelihood estimation when applied to a sample consisting of independent random variables: maximizing the product of the random variables is equivalent to maximizing the logarithm of the product, and in so doing one differentiates a sum rather than a product.

Benford's law, an empirical statistical description of the occurrence of digits in certain real-life data sources, such as heights of buildings, is based on logarithms: the probability that the first decimal digit of an item in the data sample is d (from 1 to 9) equals

log₁₀(d + 1) − log₁₀(d),

irrespective of the unit of measurement. 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.

A log-normal distribution is one whose logarithm is normally distributed.

Complexity

Quicksort sorts by choosing a pivot element (red), and sorts separately the items which are smaller than this element and the ones which are bigger.

Logarithms arise in complexity theory, a branch of computer science which studies the performance of algorithms. They are prone to occur in describing algorithms which divide a problem into two smaller ones, and join the solution of the subproblems. For example, to find a number in an sorted list, the binary search algorithm checks the middle entry and proceeds with the half before or after the middle entry if the number is still not found. Similarly, the quicksort algorithm sorts an array by dividing the array into halves and sorting these first, as shown in the animation. These algorithms typically need time that is approximately proportional to log(N) and N · log(N), respectively, where N is the length of the list.

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.) 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 change-of-base 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. 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.

Entropy

Entropy, broadly speaking a measure of (dis-)order of some system, also relies on logarithms. In thermodynamics, the entropy S of some physical system is defined by

$S=-k\sum _{i}p_{i}\ln(p_{i}).\,$

The sum is over all states i the system in question can have, p_i is the probability that the state i is attained and k is the Boltzmann constant. Similarly, 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.

Fractals

The Sierpinski triangle (at the right) is constructed by repeatedly replacing equilateral triangles by three smaller ones.

Logarithms occur in various definitions of the dimension of fractals. Fractals are geometric objects that are self-similar, i.e., that have small parts which reproduce (at least roughly) the entire global structure. The Sierpinski triangle, for example, can be covered by three copies of it having half the original size. This causes the Hausdorff dimension of this structure to be log(3)/log(2) ≈ 1.58. The idea of scaling invariance is also inherent to Benford's law mentioned above. Another logarithm-based notion of dimension is obtained by counting the number of boxes needed to cover the fractal in question.

Number theory

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

Natural logarithms are closely linked to counting prime numbers, an important topic 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

${\frac {x}{\ln(x)}},$

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

${\mbox{Li}}(x)=\int _{2}^{x}{\frac {1}{\ln(t)}}\,{\mbox{d}}t.$

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.

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.

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. 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	Semitone	Just major third	Major third	Tritone	Octave
Frequency ratio r	$2^{\frac {1}{72}}\approx 1.0097$	$2^{\frac {1}{12}}\approx 1.0595$	${\tfrac {5}{4}}=1.25$	${\begin{aligned}2^{\frac {4}{12}}&={\sqrt[{3}]{2}}\\&\approx 1.2599\end{aligned}}$	${\begin{aligned}2^{\frac {6}{12}}&={\sqrt {2}}\\&\approx 1.4142\end{aligned}}$	2
$\log _{\sqrt[{12}]{2}}(r)=12\cdot \log _{2}(r)$, i.e., corresponding number of semitones	1/6	1	≈ 3.86	4	6	12
$\log _{\sqrt[{1200}]{2}}(r)=1200\cdot \log _{2}(r)$, i.e., corresponding number of cents	16.67	100	≈ 386.31	400	600	1200

Related notions

The cologarithm of a number is the logarithm of the reciprocal of the number: colog_b(x) = log_b(1/x) = −log_b(x). 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. Both terminologies are primarily found in older books.

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.

From the perspective of pure mathematics, 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. By means of that isomorphism, the Lebesgue measure dx on R corresponds to the Haar measure dx/x on the positive reals. In complex analysis and algebraic geometry, differential forms of the form (d log(f) =) df/f are known as forms with logarithmic poles. This notion in turn gives rise to concepts such as logarithmic pairs, log terminal singularities or logarithmic geometry.

The polylogarithm is the function defined by

$\operatorname {Li} _{s}(z)=\sum _{k=1}^{\infty }{z^{k} \over k^{s}}.$

It is related to the natural logarithm by Li₁(z) = −ln(1 − z). Moreover, Li_s(1) equals the Riemann zeta function ζ(s).

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. Zech's logarithm is related to the discrete logarithm in the multiplicative group of non-zero elements of a finite field. 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, more specifically in elliptic curve cryptography.

For p-adic numbers, the logarithm can also be defined via its Taylor series, which again has radius of convergence equal to 1. This definition can be extended to all non-zero p-adic numbers. Similarly, a p-adic exponential can be defined (though its domain is smaller than in the real or complex case) and the p-adic logarithm is, once again, its inverse function.

The logarithm of a matrix is the inverse function of the matrix exponential.

History

Predecessors

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 relations such as the above product formula and also introduced logarithms in base 3 (trakacheda) and base 4 (caturthacheda).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.

John Napier

John Napier (1550–1617)

The method of logarithms was publicly propounded in 1614, in a book entitled Mirifici Logarithmorum Canonis Descriptio, by John Napier. (Joost Bürgi independently discovered logarithms; however, he did not publish his discovery until four years after Napier).

By repeated subtractions Napier calculated 10⁷(1 − 10⁻⁷)^L for L ranging from 1 to 100. The result for L=100 is approximately 0.99999 = 1 − 10⁻⁵. Napier then calculated the products of these numbers with 10⁷(1 − 10⁻⁵)^L for L from 1 to 50 and did similarly with 0.9995 ≈ (1 − 10⁻⁵)²⁰ and 0.99 ≈ 0.995²⁰. These computations, which occupied Napier for 20 years, allowed to give for any number N between 5.000.000 and 10⁷, the number L solving the equation

N = 10⁷(1 − 10⁻⁷)^L.

Napier first called L "artificial number", but later introduced the word "logarithm" to mean a number that indicates a ratio: λόγος (logos) meaning proportion, and ἀριθμός (arithmos) meaning number. In modern notation, L is approximately equal to 10⁷log_1/e(N/10⁷), because (1 − 10⁻⁷)¹⁰⁷ is approximately 1/e. However, the number e and the modern definition for logarithms were developed about 100 years later, by Euler in 1728.

The invention of logarithms was quickly met with acclaim in many countries. The work of Cavalieri (Italy), Wingate (France), Fengzuo (China), and Kepler's Chilias logarithmorum (Germany) helped spread the concept further.

Tables of logarithms and historical applications

The 1797 Encyclopædia Britannica entry "logarithms".

Logarithms contributed to the advance of science, and especially of astronomy. Prior to their invention, the more involved method of prosthaphaeresis, which relied on trigonometric identities, was used to approximately calculate products. Until the advent of calculators and computers, logarithms were used constantly in surveying, celestial navigation, and other branches of practical mathematics. Laplace called logarithms

[a]n admirable artifice which, by reducing to a few days the labour of many months, doubles the life of the astronomer, and spares him the errors and disgust inseparable from long calculations.

The calculation of products using logarithms stakes on the following formula:

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

The base b is fixed. Many tables use the common logarithm, i.e., b = 10. The logarithms log_bc and log_bd are looked up in a table of logarithms. Calculating the sum of the logarithms is an easy operation. Finally, raising b to the power of this sum (historically called antilogarithm) is again done by looking up a table. For manual calculations that demand any appreciable precision, this process, requiring three lookups and a sum, is much faster than performing the multiplication. The precision of the approximation can be increased by interpolating between table entries.

Calculation of powers are reduced to multiplications and look-ups by

c^d = b^{(log_bc) · d}.

Divisions and roots are also covered by these two techniques since

${\frac {c}{d}}=c\cdot d^{-1}=b^{\log _{b}c-\log _{b}d}$

and

${\sqrt[{d}]{c}}=c^{\frac {1}{d}}$.

For different needs, tables ranging from small handbooks to multi-volume editions have been compiled. The following table lists the major tables of logarithms compiled in history:

Year	Author	Range	Decimal places	Note
1617	Henry Briggs	1–1000	8
1624	Henry Briggs Arithmetica Logarithmica	1–20,000, 90,000–100,000	14
1628	Adriaan Vlacq	20,000–90,000	10	contained only 603 errors
1792–94	Gaspard de Prony Tables du Cadastre	1–100,000 and 100,000–200,000	19 and 24, respectively	"seventeen enormous folios", never published
1794	Jurij Vega Thesaurus Logarithmorum Completus (Leipzig)			corrected edition of Vlacq's work
1795	François Callet (Paris)	100,000–108,000	7
1871	Sang	1–200,000	7

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

Another critical application was the slide rule. The non-sliding logarithmic scale or Gunter's rule was invented by Edmund Gunter shortly after Napier's invention. This design was improved by William Oughtred into the slide rule: a pair of logarithmic scales movable with respect to each other. The slide rule was an essential calculating tool for engineers and scientists until the 1970s since it allows much faster computation than techniques based on tables. The slide rule provides somewhat less precision than typical table-based calculations, but enough for many types of engineering.

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.

Page of a 1912 logarithm table.

See also

• List of logarithm topics
• Logit
• Lyapunov exponents
• Logarithmic number system
• Logarithmic differentiation
• Log return

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. In fact, the notation was invented by a mathematician Irving Stringham.
^ b: The same series holds the principal value of the complex logarithm for complex numbers z satisfying |z − 1| < 1.
^ c: For rules concerning exponentiation such as b⁻¹ = 1/b, b^{m + n} = b^m · bⁿ, see exponentiation or for an elementary treatise.
^ d: See radian for the conversion between 2π and 360 degrees.

External links

• Educational video on logarithms, retrieved 12 October 2010
• Translation of Napier's work on logarithms, retrieved 12 October 2010