Logarithm: Difference between revisions

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 the number.

For example, the logarithm of 1000 to the base 10 is 3, because 3 is how many 10s you must multiply to get 1000: thus 10 × 10 × 10 = 1000; the base 2 logarithm of 32 is 5 because 5 is how many 2s one must multiply to get 32: thus 2 × 2 × 2 × 2 × 2 = 32. In the language of exponents: 10³ = 1000, so log_{101000  = 3, and 2⁵ = 32, so log232 = 5.}

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,

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

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

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

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.

When x and b are restricted to positive real numbers, log_b(x) is a unique real number. The magnitude of the base b must be neither 0 nor 1; the base used is typically 10, e, or 2. Logarithms are defined for real numbers and for complex numbers. ^[1][2]

The major property of logarithms is that they map multiplication to addition. This ability stems from the following identity:

${\displaystyle b^{x}\times b^{y}=b^{x+y}\ ,}$

which by taking logarithms becomes

${\displaystyle \log _{b}\left(b^{x}\times b^{y}\right)=\log _{b}\left(b^{x+y}\right)}$ ${\displaystyle \ =x+y=\log _{b}\left(b^{x}\right)+\log _{b}\left(b^{y}\right).\ }$

For example,

${\displaystyle 4=2^{2}\,\Rightarrow \,\log _{2}(4)=2,\,}$

${\displaystyle 8=2^{3}\,\Rightarrow \,\log _{2}(8)=3,\,}$

${\displaystyle \log _{2}(32)=\log _{2}(4\times 8)=\log _{2}(4)+\log _{2}(8)=2+3=5.\,}$

A related property is reduction of exponentiation to multiplication. Using the identity:

${\displaystyle c=b^{\log _{b}(c)}\ ,}$

it follows that c to the power p (exponentiation) is:

${\displaystyle c^{p}=\left(b^{\log _{b}(c)}\right)^{p}=b^{p\log _{b}(c)}\ ,}$

or, taking logarithms:

${\displaystyle \log _{b}\left(c^{p}\right)=p\log _{b}(c).\,}$

In words, to raise a number to a power p, find the logarithm of the number and multiply it by p. The exponentiated value is then the inverse logarithm of this product; that is, number to power = b^product.

For example,

${\displaystyle \log _{2}(64)=\log _{2}(4^{3})=3\log _{2}(4)=6.\,}$

Besides reducing multiplication operations to addition, and exponentiation to multiplication, logarithms reduce division to subtraction, and roots to division. For example,

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

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

Logarithms make lengthy numerical operations easier to perform. The whole process is made easy by using tables of logarithms, or a slide rule, antiquated now that calculators are available. Although the above practical advantages are not important for numerical work today, they are used in graphical analysis (see Bode plot).

Though logarithms have been traditionally thought of as arithmetic sequences of numbers corresponding to geometric sequences of other (positive real) numbers, as in the 1797 Britannica definition, they are also the result of applying an analytic function. The function can therefore be meaningfully extended to complex numbers.

The function log_b(x) depends on both b and x, but the term logarithm function (or logarithmic function) in standard usage refers to a function of the form log_b(x) in which the base b is fixed and so the only argument is x. Thus there is one logarithm function for each value of the base b (which must be positive and must differ from 1). Viewed in this way, the base-b logarithm function is the inverse function of the exponential function b^x. The word "logarithm" is often used to refer to a logarithm function itself as well as to particular values of this function.

Only positive real numbers have real-valued logarithms. The logarithm function can be extended to the complex logarithm, which applies to negative and complex numbers and yields a complex number. The value is not unique though, since for example ${\displaystyle e^{2\pi i}=e^{0}=1}$ which implies that both ${\displaystyle 2\pi i}$ and 0 are equally valid logarithms to base e of 1.

When z is a complex number, say z = x + iy where x and y are real, the logarithm of z is found by putting z in polar form that is, z = re^iθ = r(cos θ + i sin θ), where ${\displaystyle r=|z|={\sqrt {x^{2}+y^{2}}}}$ and θ = arg(z) is any angle such that x = r cos θ and y = r sin θ. The function arg is a multi-valued function.

If the base of the logarithm is chosen as e ^[3], that is, using log_e (denoted by ln and called the natural logarithm), the complex logarithm is:

${\displaystyle \log(z)=\ln |z|+i\arg(z)=\ln(r)+i\left(\theta +2\pi k\right)}$

which is, just like arg, also a multi-valued function. The principal value of the logarithm, Log (denoted by a capital first letter), is a single-valued function and is defined as

${\displaystyle {\text{Log}}(z)=\ln |z|+i{\text{Arg}}(z)=\ln(r)+i\varphi }$

where ${\displaystyle \varphi }$ is the (only) value in the range ${\displaystyle (-\pi ,\ \pi ]}$ which is ${\displaystyle \equiv \theta {\pmod {2\pi }}\ .}$

The function Arg is the principal argument. It is a single-valued function and defined as the branch of ${\displaystyle \arg }$ in which the values are in the range ${\displaystyle (-\pi ,\ \pi ],}$ leaving a branch cut at the negative reals. 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 number r is:

${\displaystyle {\text{Log}}(r)=\ln |r|+i\pi \ .}$

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).

Note that log(z^p) is not in general the same as p log(z); see failure of power and logarithm identities.

From the pure mathematical perspective, the identity

${\displaystyle \log(cd)=\log(c)+\log(d)\,}$

is fundamental in two senses. First, the remaining three arithmetic properties can be derived from it. Furthermore, it expresses an isomorphism between the multiplicative group of the positive real numbers and the additive group of all the reals.

Logarithmic functions are the only continuous isomorphisms from the multiplicative group of positive real numbers to the additive group of real numbers.

The most widely used bases for logarithms are 10, the mathematical constant e ≈ 2.71828... and 2. When "log" is written without a base (b missing from log_b), the intent can usually be determined from context:

• natural logarithm (log_e, ln, log, or Ln) in mathematical analysis, statistics, economics and some engineering fields. The reasons to consider e the natural base for logarithms, though perhaps not obvious, are numerous and compelling. (Euler's identity is important to fields that deal with cyclic components.)
• common logarithm (log₁₀ or simply log; sometimes lg) 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. Use is historically grounded. (see dB) Also, the approximation 2¹⁰≈10³ leads to the approximations 3 dB per octave (power doubling) – a useful result that occurs with the use of log₁₀.
• binary logarithm (log₂; sometimes lg, lb, or ld), in computer science and information theory. 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 lack the intuitive interpretation, although the units have equivalent function, differing only in scale.
• indefinite logarithm (Log or [log ] or simply log) when the base is irrelevant, e.g. in complexity theory when describing the asymptotic behavior of algorithms in big O notation, which describes the character of the algorithm, i.e. "the behavior is logarithmic", not the exact measure of performance of the algorithm in a given situation.

To avoid confusion, it is best to specify the base if there is any chance of misinterpretation.

The notation "ln(x)" invariably means log_e(x), i.e., the natural logarithm of x. However, if, as in "log(x)", the base is not given explicitly, it may be understood implicitly by discipline:

• Mathematicians understand "log(x)" to mean log_e(x). Calculus textbooks^{[citation needed]} will occasionally write "log(x)" to represent "log₁₀(x)".

• Many engineers, biologists, astronomers, and some others write only "ln(x)" or "log_e(x)" when they mean the natural logarithm of x, and take "log(x)" to mean log₁₀(x) or, in computer science, log₂(x).

• On most calculators, the LOG button is log₁₀(x) and LN is log_e(x).

• In most commonly used computer programming languages, including C, C++, Java, Haskell, Fortran, Python, Ruby, and BASIC, the "log" function returns the natural logarithm. The base-10 function, if it is available, is generally "log10."

• Some people use Log(x) (capital L) to mean log₁₀(x), and use log(x) with a lowercase l to mean log_e(x).

• The notation Log(x) is also used by mathematicians to denote the principal branch of the (natural) logarithm function.

• In some European countries, a frequently used notation is ^blog(x) instead of log_b(x).^[4]

This chaos, historically, originates from the fact that the natural logarithm has nice mathematical properties (such as its derivative being 1/x, and having a simple definition), while the base 10 logarithms, or decimal logarithms, were more convenient for speeding calculations (back when they were used for that purpose). Thus natural logarithms were only extensively used in fields like calculus while decimal logarithms were widely used elsewhere.

As recently as 1984, Paul Halmos in his "automathography" I Want to Be a Mathematician heaped contempt on what he considered the childish "ln" notation, which he said no mathematician had ever used. The notation was in fact invented in 1893 by Irving Stringham, professor of mathematics at Berkeley.^{[5][citation needed]}

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 log, and lb(x) for the binary log.^[6] In Russian literature, the notation lg(x) is also generally used for the base 10 logarithm.^[7] 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.

The clear advice of the United States Department of Commerce National Institute of Standards and Technology is to follow the ISO standard Mathematical signs and symbols for use in physical sciences and technology, ISO 31-11:1992, which suggests these notations:^[8]

• The notation "ln(x)" means log_e(x);
• The notation "lg(x)" means log₁₀(x);
• The notation "lb(x)" means log₂(x).

As the difference between logarithms to different bases is one of scale, it is possible to consider all logarithm functions to be the same, merely giving the answer in different units, such as dB, neper, bits, decades, etc.; see the section Science and engineering below. 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.

While there are several useful identities, the most important for calculator use lets one find logarithms with bases other than those built into the calculator (usually log_e and log₁₀). To find a logarithm with base b, using any other base k:

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

This is because the definition of logarithm says that

${\displaystyle x=b^{a}\iff \log _{b}{x}=a\,}$

but we can also get a by using the base k logarithm and then get

${\displaystyle \log _{k}{x}=\log _{k}{b^{a}}=a\log _{k}{b}\iff {\frac {\log _{k}{x}}{\log _{k}{b}}}=a}$

with b ≠ 1, because log_k 1 = 0. Any number to the power 0 is equal to 1.

Moreover, this result implies that all logarithm functions (whatever the base) are similar to each other.

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.

Various quantities in science are expressed as logarithms of other quantities; see logarithmic scale for an explanation and a more complete list.

• 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. (This is a result of the equilibrium constant, the product of the concentration of hydronium ions and hydroxyl ions, in water solutions being 10⁻¹⁴ M².)
• The bel (symbol B) 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. The Bel is named after telecommunications pioneer Alexander Graham Bell. The decibel (dB), equal to 0.1 bel, is more commonly used. The neper is a similar unit which uses the natural logarithm of a ratio.
• The Richter scale measures earthquake intensity on a base-10 logarithmic scale.
• In spectrometry and optics, the absorbance unit used to measure optical density is equivalent to −1 B.
• In astronomy, the apparent magnitude measures the brightness of stars logarithmically, since the eye also responds logarithmically to brightness.
• In psychophysics, the Weber–Fechner law proposes a logarithmic relationship between stimulus and sensation.
• 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.
• In geometry the logarithm is used to form the metric for the half-plane model of hyperbolic geometry.
• Many types of engineering and scientific data are typically graphed on log-log or semilog axes, in order to most clearly show the form of the data.
• 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.
• 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).

One way of defining the exponential function e^x, also written as exp(x), is as the inverse of the natural logarithm. It is positive for every real argument x.

The operation of "raising b to a power p" for positive arguments b and all real exponents p is defined by

${\displaystyle b^{p}=\left(e^{\ln b}\right)^{p}=e^{p\ln b}.\,}$

Logarithms can be used to replace difficult operations on numbers by easier operations on their logs (in any base), as the following table summarizes. In the table, upper-case variables represent logs of corresponding lower-case variables:

Operation with numbers	Operation with exponents	Logarithmic identity
${\displaystyle \!\,cd}$	${\displaystyle \!\,C+D}$	${\displaystyle \!\,\log(cd)=\log(c)+\log(d)}$
${\displaystyle \!\,{\frac {c}{d}}}$	${\displaystyle \!\,C-D}$	${\displaystyle \!\,\log \left({\frac {c}{d}}\right)=\log(c)-\log(d)}$
${\displaystyle \!\,c^{d}}$	${\displaystyle \!\,Cd}$	${\displaystyle \!\,\log(c^{d})=d\log(c)}$
${\displaystyle \!\,{\sqrt[{d}]{c}}}$	${\displaystyle \!\,{\frac {C}{d}}}$	${\displaystyle \!\,\log({\sqrt[{d}]{c}})={\frac {\log(c)}{d}}}$

These arithmetic properties of logarithms make such calculations much faster. 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.

As an example, to approximate the product of two numbers one can look up their logarithms in a table, add them, and, using the table again, proceed from that sum to its antilogarithm, which is the desired product. The precision of the approximation can be increased by interpolating between table entries. 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. Similarly, to approximate a power c^d one can look up log c in the table, look up the log of that, and add to it the log of d; roots can be approximated in much the same way.

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, an essential calculating tool for engineers. Many of the powerful capabilities of the slide rule derive from a clever but simple design that relies on the arithmetic properties of logarithms. The slide rule allows computation much faster still than the techniques based on tables, but provides much less precision, although slide rule operations can be chained to calculate answers to any arbitrary precision.

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.^[9]

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.^[10] The notation still appears in some modern books, and is still used in some situations. For example, certain electronic circuit components are known as antilog amplifiers.^[11]

The Lambert W function is the inverse function of ƒ(w) = we^w.

The natural logarithm of a positive number x can be defined as

${\displaystyle \ln :\mathbf {R} _{+}^{*}\longmapsto \mathbf {R} ,\quad \ln(x):=\int _{1}^{x}{\frac {dt}{t}}.}$

This function is also commonly denoted by log.

This definition satisfies the usual properties of a logarithm. For example, it can be shown as follows that ln(x^r) = r ln(x). To see this, consider the definition ${\displaystyle \ln(x^{r}):=\int _{1}^{x^{r}}{\frac {1}{t}}\,dt}$ and the change of variable u := t^1/r. Then, by the integration by substitution theorem:

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

Likewise, it can be shown that this function verifies the property ln(xy) = ln(x) + ln(y) using

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

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

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

as desired.

Using the last two properties, the rule ln(x / y) = ln(x) − ln(y) can be proved:

${\displaystyle \ln \left({\frac {x}{y}}\right)=\ln(xy^{-1})=\ln(x)+\ln(y^{-1})=\ln(x)-\ln(y).\,}$

The derivative of the natural logarithm function is

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

By applying the change-of-base rule, the derivative for other bases is

${\displaystyle {\frac {d}{dx}}\log _{b}(x)={\frac {d}{dx}}{\frac {\ln(x)}{\ln(b)}}={\frac {1}{x\ln(b)}}={\frac {\log _{b}(e)}{x}}.}$

The antiderivative of the natural logarithm ln(x) is

${\displaystyle \int \ln(x)\,dx=x\ln(x)-x+C,}$

and so the antiderivative of the logarithm for other bases is

${\displaystyle \int \log _{b}(x)\,dx=x\log _{b}(x)-{\frac {x}{\ln(b)}}+C=x\log _{b}\left({\frac {x}{e}}\right)+C.}$

See also: Table of limits, list of integrals of logarithmic functions.

There are several series for calculating natural logarithms.^[12] The simplest, though inefficient, is:

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

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

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

Integrate both sides to obtain

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

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

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

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

A more efficient series is

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

for z with positive real part.

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

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

Subtracting, we get

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

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

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

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.

For example, applying this series to

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

we get

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

and thus

${\displaystyle \ln(1.2222222\dots )={\frac {2}{10}}\left(1+{\frac {1}{3\cdot 100}}+{\frac {1}{5\cdot 10000}}+{\frac {1}{7\cdot 1000000}}+\cdots \right)}$

${\displaystyle =0.2\cdot (1.0000000\dots +0.0033333\dots +0.0000200\dots +0.0000001\dots +\cdots )}$

${\displaystyle =0.2\cdot 1.0033535\dots =0.2006707\dots }$

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

For any other base b, we use

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

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

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

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

Many computer languages use log(x) for the natural logarithm, while the common log 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

${\displaystyle x=m2^{n}.\,}$ (Sometimes a base other than 2 is used.)

Therefore

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

Thus, instead of computing ${\displaystyle \ln(x)}$ we compute ${\displaystyle \ln(m)}$ for some m such that 1 ≤ m <  2. Having m in this range means that the value ${\displaystyle \textstyle u={\frac {m-1}{m+1}}}$ is always in the range ${\displaystyle \textstyle 0\leq u<{\frac {1}{3}}}$. Some machines use the significand in the range ${\displaystyle 0.5\leq m<1}$ and in that case the value for u will be in the range ${\displaystyle \textstyle -{\frac {1}{3}}<u\leq 0}$ 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)

The ordinary logarithm of positive reals generalizes to negative and complex arguments, though it is a multivalued function that needs a branch cut terminating at the branch point at 0 to make an ordinary function or principal branch. The logarithm (to base e) of a complex number z is the complex number ln(|z|) + i arg(z), where |z| is the modulus of z, arg(z) is the argument, and i is the imaginary unit; see complex logarithm for details.

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.

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

It is possible to take the logarithm of a quaternions and octonions.

A double logarithm, ${\displaystyle \ln(\ln(x))}$, is the inverse function of the double exponential function. A super-logarithm 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.

For each positive b not equal to 1, the function log_b  (x) is an isomorphism from the group of positive real numbers under multiplication to the group of (all) real numbers under addition. They are the only such isomorphisms that are continuous. The logarithm function can be extended to a Haar measure in the topological group of positive real numbers under multiplication.

The method of logarithms was first publicly propounded in 1614, in a book entitled Mirifici Logarithmorum Canonis Descriptio, by John Napier, Baron of Merchiston, in Scotland.^[13] (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.^[14]

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.^[6]

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."^[15] 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 1700s. 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." ^[16] Cubic interpolation could be used to find the logarithm of any number to a similar accuracy.

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