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 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,

${\text{ if }}x=b^{y},{\text{ then }}y=\log _{b}(x)\,.$

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:

$\log(xy)=\log x+\log y\,.$

That is, the logarithm of the product of two numbers is the sum of 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 positive real number x with respect to another positive real number b, where b is not equal to 0 or 1, is defined to be the real number s such that

b^s = x,

that is, the s-th power of b must equal x. To see that s exists and is unique, note that if 0 < b < 1 then b^s maps the real numbers to the positive real numbers and is a strictly decreasing function of s, so b^s = x has one and only one solution if x is real and positive. Similarly, there is one and only one solution if b > 1, because b^s is then a strictly increasing function of s.

The logarithm s is denoted log_b(x). (Some European countries write ^blog(x) instead.) The number b is referred to as the base. For b = 2, for example, this means

log₂(8) = 3,

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

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

since

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

Identities

Main article: Logarithmic Identities

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

Logarithm of products

First of all, 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

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 powers

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

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)}.

thus, by taking logarithms:

log_b(x^p) = p log_b(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,

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

$\log _{2}({\sqrt[{3}]{4}})={\frac {1}{3}}\log _{2}(4)={\frac {2}{3}}.\,$

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:

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

In fact, 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 all logarithm functions (whatever the base b) are similar to each other.

Bases

While the definition of logarithm applies to any positive real number b (0 and 1 are excluded, though), a few particular choices for b are more commonly used. These are b = 10, b = e, the mathematical constant e ≈ 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). The common logarithm of a number x tells how many numerical digits x has: when

e ≤ log₁₀(x) < f,

with two integers e and f, x has e decimal digits. Since we write numbers in base 10, mental math is thus easier with the common log, 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, log_e(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 log_e(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. In fact, the notation was invented by a mathematician, Irving Stringham, professor of mathematics at University of California, Berkeley, in 1893.

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 log₂(n) bits to store the integer n. Likewise, a binary search through a list of size n takes log₂(n) steps. Properties like this come up repeatedly in these domains.

Implicit bases

Instead of writing log_b(x), it is common to omit the base, log(x), when intended base can be determined from context. In mathematics and many programming languages , "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 log_e(x). The International Organization for Standardization (ISO 31-11) suggests the notations "ln(x)" ("lg(x)", "lb(x)") for log_e(x) (log₁₀(x) and log₂(x), respectively).

The base, b, used by the supplied logarithm function can be explicitly determined using the following identity (subject to the inherent computational accuracy errors).

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

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

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

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 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.

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 of a negative or complex number

Main article: 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. Given any non-zero complex number z = x + iy, the intent is—as with the natural logarithm of real numbers above— to find a number a such that

e^a = z.

Polar form of complex numbers

To solve the equation, it is convenient to express z in polar form:

z = re^iθ = r(cos(θ) + i sin(θ)),

where $r=|z|={\sqrt {x^{2}+y^{2}}}$ is the absolute value of z and θ = arg(z) the argument of z, that is, is any angle such that x = rcos(θ) and y = r sin(θ). Then,

a = ln(r) + i θ

is such that the a-th power of e equals z. Hence a qualifies being called logarithm of z. However, the argument θ is not unique. In fact, adding θ' = θ + 2πi is also an argument of z, and

a' = ln(r) + i θ'

also satisfies e^a' = z. This is a consequence of

e^2πi = 1.

It can be shown that any solution a to e^a = z is necessarily of the form a = ln(r) + i θ + 2nπi, where n is an arbitrary integer.

The function

log z := ln (|z|) + i arg(z) = ln(r) + i (θ + 2πn), n ∈ Z

is therefore multi-valued. The ambiguity can be fixed by requiring -π < θ ≤ π, the so-called principal argument, denoted Arg. The principal value of the logarithm, Log (denoted by a capital first letter), is a single-valued function and is defined as

Log (z) = ln (|z|) + i Arg (z) = ln(r) + i φ,

where φ is the (only) value in the range (-π, π]</math> which is θ plus some integer multiple of 2π. 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 principal value of the logarithm of a negative real number x is:

Log (x) = ln|x| + i π.

For a base b other than e the complex logarithm log_b(z) can be defined as ln(z)/ln(b), the principal value of which is given by the principal values of ln(z) and ln(b).

Analogous formula for principal values of logarithm of products and powers for complex numbers do in general not hold.

Uses and occurrences

Logarithms are useful in solving equations in which exponents are unknown. They have simple derivatives, so they are often used in the solution of integrals. The logarithm is one of three closely related functions. In the equation bⁿ = x, b can be determined with radicals, n with logarithms, and x with exponentials. See logarithmic identities for several rules governing the logarithm functions.

Calculation of products, powers and roots

Logarithms can be used to reduce multiplications and exponentions, which are difficult to perform by hand, to additions. This was the historical motivation for logarithms. The use of logarithms was an essential skill until electronic 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.

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: ${\frac {c}{d}}=c\cdot d^{-1}=b^{\log _{b}c-\log _{b}d}.$ Moreover,

c^d = b^{(log_bc) · 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 ${\sqrt[{d}]{c}}$ can be calculated this way, too, since ${\sqrt[{d}]{c}}=c^{\frac {1}{d}}$.

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

Main 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 say. Considering not the quantity itself, but its logarithm 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 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, 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 · b^x 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.

Psychology

In psychophysics, the Weber–Fechner law proposes a logarithmic relationship between stimulus and sensation (though the more modern Stevens' power law is typically more accurate). According to the logarithmic responsiveness of the eye to brightness, the apparent magnitude measures the brightness of stars logarithmically.

Mathematically untrained indivudals 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) as in comparison between American and indigene cultures.

Information and complexity

In computer science, logarithms often appear in bounds for computational complexity. For example, to sort N items using comparison can require 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 2^k distinct values, so any natural number N can be represented in no more than (log₂ N) + 1 bits.

Similarly, in information theory logarithms are used as 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.

Mathematics

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

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.

Music

Musical intervals are measured logarithmically as semitones. The interval between two notes in semitones is the base-2^1/12 logarithm of the frequency ratio (or equivalently, 12 times the base-2 logarithm). Fractional semitones are used for non-equal temperaments. Especially to measure deviations from the equal tempered scale, 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). In MIDI, notes are numbered on the semitone scale (logarithmic absolute nominal pitch with middle C at 60). For microtuning to other tuning systems, a logarithmic scale is defined filling in the ranges between the semitones of the equal tempered scale in a compatible way. This scale corresponds to the note numbers for whole semitones. (see microtuning in MIDI).

Analytic properties

Further information: List of integrals of logarithmic functions

The expression log_b(x) depends on both b and x, but the term logarith 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). This function is continuous (it does not "jump"), i.e., the logarithm of x changes only little when x varies only little. It approaches minus infinity when x tends to zero (while staying positive) and (plus) infinity when x grows to infinity: $\lim _{x\searrow 0}\log _{b}x=-\infty ,\,\lim _{x\rightarrow \infty }\log _{b}x=\infty$

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

Alternative definition via integrals

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. The natural logarithm of a positive number x can be alternatively defined by means of calculus:

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

the integral of 1/t dt from 1 to x. It can be shown that this formula yields the same results as the definition above. Consequently, the integral-based definition satisfies the above laws. However, they can also be deduced independently. For example, ln(x^r) = r ln(x) follows from the change of variable u := t^1/r. Integration by substitution yields

$\ln(x^{r})=\int _{1}^{x^{r}}{\frac {dt}{t}}=\int _{1}^{x}{\frac {1}{u^{r}}}\left(ru^{r-1}\,du\right)=r\int _{1}^{x}{\frac {1}{u}}\,du=r\ln(x).$

The property ln(xy) = ln(x) + ln(y) is deduced in a similar way:

$\ln(xy):=\int _{1}^{xy}{\frac {1}{t}}\,dt=\int _{1}^{x}{\frac {1}{t}}\,dt+\int _{x}^{xy}{\frac {1}{t}}\,dt=\ln(x)+\int _{x}^{xy}{\frac {1}{t}}\,dt$

Using the change of variable u = t/x in the last integral yields

$\ln(xy)=\ln(x)+\int _{1}^{y}{\frac {1}{u}}\,du=\ln(x)+\ln(y)$

as desired.

Derivative and antiderivative

As follows from the above integral definition and the fundamental theorem of calculus, the derivative of the natural logarithm function is

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

This implies that 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

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

Series for calculating the natural logarithm

Basic series

There are several series for calculating natural logarithms. The simplest, though inefficient, is

$\ln(z)=\sum _{n=1}^{\infty }-{\frac {(1-z)^{n}}{n}}{\text{ when }}|1-z|<1.$

To derive this series, start with (|x| < 1)

${\frac {1}{1-x}}=1+x+x^{2}+x^{3}+\cdots .$

Integrate both sides to obtain

${\begin{aligned}-\ln(1-x)&=x+{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}+{\frac {x^{4}}{4}}+\cdots \\[6pt]\ln(1-x)&=-x-{\frac {x^{2}}{2}}-{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}-\cdots .\end{aligned}}$

Letting z = 1 − x and thus x = 1 − z, we get

${\begin{aligned}\ln z&=-(1-z)-{\frac {(1-z)^{2}}{2}}-{\frac {(1-z)^{3}}{3}}-{\frac {(1-z)^{4}}{4}}+\cdots \\[6pt]&=(z-1)-{\frac {(z-1)^{2}}{2}}+{\frac {(z-1)^{3}}{3}}-{\frac {(z-1)^{4}}{4}}+\cdots .\end{aligned}}$

The last expression above is the familiar Taylor series.

More efficient series

A more efficient series is

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

for z with positive real part.

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

$\ln(1+x)=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}+\cdots .$

Subtracting, we get

$\ln {\frac {1+x}{1-x}}=\ln(1+x)-\ln(1-x)=2x+2{\frac {x^{3}}{3}}+2{\frac {x^{5}}{5}}+\cdots .$

Letting $z={\frac {1+x}{1-x}}\!$ and thus $x={\frac {z-1}{z+1}}\!$, we get

$\ln z=2\left({\frac {z-1}{z+1}}+{\frac {1}{3}}{\left({\frac {z-1}{z+1}}\right)}^{3}+{\frac {1}{5}}{\left({\frac {z-1}{z+1}}\right)}^{5}+\cdots \right).$

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 × 10^b, 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.

Example

For example, applying this series to

$z={\frac {11}{9}},$

we get

${\frac {z-1}{z+1}}={\frac {{\frac {11}{9}}-1}{{\frac {11}{9}}+1}}={\frac {1}{10}},$

and thus

${\begin{aligned}&{}\quad \ln(1.2222222\dots )={\frac {2}{10}}\left(1+{\frac {1}{3\cdot 100}}+{\frac {1}{5\cdot 10000}}+{\frac {1}{7\cdot 1000000}}+\cdots \right)\\&=0.2\cdot (1.0000000\ldots +0.0033333\ldots +0.0000200\ldots +0.0000001\ldots +\cdots )\\&=0.2\cdot 1.0033535\ldots =0.2006707\dots \end{aligned}}$

where we factored 2/10 out of the sum in the first line.

For any other base b, we use

$\log _{b}(x)={\frac {\ln(x)}{\ln(b)}}.$

About convergence

The above series for ln(1 − x) converges for all complex number |x| ≤ 1, x ≠ 1. In fact, as seen by the ratio test, it has radius of convergence equal to 1, therefore converges absolutely on every disk B(0, r) with radius r < 1. Moreover, it converges uniformly on every nibbled disk $\scriptstyle {\overline {B(0,1)}}\setminus B(1,\delta )$, with δ > 0. This follows at once from the algebraic identity:

$(1-x)\sum _{n=1}^{m}{\frac {x^{n}}{n}}=x-\sum _{n=2}^{m}{\frac {x^{n}}{n(n-1)}}-{\frac {x^{m+1}}{m}},$

just observing that the right-hand side is uniformly convergent on the whole closed unit disk.

Computers

Many computer languages use log(x) as a function for the natural logarithm, while the common logarithm is typically denoted log10(x). The argument and return values are typically a floating point (or double precision) data type.

As the argument is floating point, it can be useful to consider the following:

A floating point value x is represented by a significand m and exponent n to form

$x=m2^{n}.\,$

(Sometimes a base other than 2 is used.)

Therefore

$\ln(x)=\ln(m)+n\ln(2).\,$

Thus, instead of computing ln(x) we compute ln(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.

To compute a base-2 logarithm on a number between 1 and 2 in an alternate way, square it repeatedly. Every time it goes over 2, divide it by 2 and write a "1" bit, else just write a "0" bit. This is because squaring doubles the logarithm of a number.

The characteristic (integer part of the logarithm) to base 2 of an unsigned integer is given by the position of the leftmost bit, and can be computed in O(n) steps using the following algorithm:

int log2(unsigned int x) {
  int r = 0;
  while ((x >> r) != 0) {
    r++;
  }
  return r-1; // returns -1 for x==0, floor(log2(x)) otherwise
}

However, it can also be computed in O(log n) steps by trying to shift by powers of 2 and checking that the result stays nonzero: for example, first >> 16, then >> 8, ... (each step reveals one bit of the result).

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.

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. 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. Today's applications of antilogarithms include certain electronic circuit components known as antilog amplifiers, 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

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

For s = 1, it is related to the logarithm via Li₁(z) = −ln(1 − z). Forz = 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. The logarithm function can be extended to a Haar 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.

Logarithms can be defined for quaternions and octonions and matrices. 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

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, (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.

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: Template:Polytonic (logos) meaning proportion, and Template:Polytonic (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.

Tables of logarithms

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

Prior to the advent of computers and calculators, using logarithms meant using tables of logarithms, which had to be created manually. Base-10 logarithms are useful in computations when electronic means are not available. See common logarithm for details, including the use of characteristics and mantissas of common (i.e., base-10) logarithms.

In 1617, Henry Briggs published the first installment of his own table of common logarithms, containing the logarithms of all integers below 1000 to eight decimal places. This he followed, in 1624, by his Arithmetica Logarithmica, containing the logarithms of all integers from 1 to 20,000 and from 90,000 to 100,000 to fourteen places of decimals, together with a learned introduction, in which the theory and use of logarithms are fully developed. The interval from 20,000 to 90,000 was filled up by Adriaan Vlacq, a Dutch mathematician; but in his table, which appeared in 1628, the logarithms were given to only ten places of decimals.

Vlacq's table was later found to contain 603 errors, but "this cannot be regarded as a great number, when it is considered that the table was the result of an original calculation, and that more than 2,100,000 printed figures are liable to error." An edition of Vlacq's work, containing many corrections, was issued at Leipzig in 1794 under the title Thesaurus Logarithmorum Completus by Jurij Vega.

François Callet's seven-place table (Paris, 1795), instead of stopping at 100,000, gave the eight-place logarithms of the numbers between 100,000 and 108,000, in order to diminish the errors of interpolation, which were greatest in the early part of the table; and this addition was generally included in seven-place tables. The only important published extension of Vlacq's table was made by Mr. Sang in 1871, whose table contained the seven-place logarithms of all numbers below 200,000.

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

Besides the tables mentioned above, a great collection, called Tables du Cadastre, was constructed under the direction of Gaspard de Prony, by an original computation, under the auspices of the French republican government of the 1790s. This work, which contained the logarithms of all numbers up to 100,000 to nineteen places, and of the numbers between 100,000 and 200,000 to twenty-four places, exists only in manuscript, "in seventeen enormous folios," at the Observatory of Paris. It was begun in 1792; and "the whole of the calculations, which to secure greater accuracy were performed in duplicate, and the two manuscripts subsequently collated with care, were completed in the short space of two years." Cubic interpolation could be used to find the logarithm of any number to a similar accuracy.

See also

• List of logarithm topics
• List of logarithmic identities
• Imaginary-base logarithm
• Zech's logarithms
• Log-normal distribution
• Equal temperament
• Logarithmic number system
• Entropy (information theory)

References

• Dehaene S, Izard V, Spelke E, Pica P. 2008. Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures. Science 320:1217-1220. doi:10.1126/science.1156540

External links

• Explaining Logarithms
• Logarithms on geoset.info
• Weisstein, Eric W. "Logarithm". MathWorld.
• Jost Burgi, Swiss Inventor of Logarithms
• Translation of Napier's work on logarithms
• Algorithm for determining Log values for any base