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 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.

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 the number.

For example, the logarithm of 1000 to the base 10 is 3, because 10 raised to the power of 3 is 1000; the base 2 logarithm of 32 is 5 because 2 to the power 5 is 32.

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,

 if b^y = x, then y = log_b(x)

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 complex calculations was a significant motivation in their original development.

Properties of the logarithm

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.

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

b^x Ž b^y = b^x+y

which by taking logarithms becomes

log_b(b^x Ž b^y) = log_b(b^x+y) = x + y = log_b(b^x) + log_b(b^y)

For example,

4 = 2² Þ log₂(4) = 2,

8 = 2³ Þ log₂(8) = 3,

Log₂(32) = log₂(4Ž8) = log₂(4) + log₂(8) = 2 + 3 = 5

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

log_b(c^p) = 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,

Log₂(64) = log₂(4³) = 3log₂(4) = 6

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

log₂(16) = log₂(⁶⁴/₄) = log₂(64) - log₂(4) = 6 – 2 = 4,

log₂(4^1/3) = ¹/₃ log₂(4) = 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

The logarithm as a function

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.

Logarithm of a negative or complex number

There is no real valued logarithm for negative or complex numbers. The logarithm function can be extended to the complex logarithm. which does apply to these cases. The value is not unique though, since e^2πi = e⁰ = 1 then both 2πi and 0 are equally valid logarithms to base e of 1.

When z is a complex number, say z = x + i y, the logarithm of z is found by putting z in polar form that is, z = re^iθ = r cos(θ) + i r sin(θ), where r = |z| = sqrt(x² + y²) and θ = arg(z) is any angle such that x = r cos θ and y = r sin θ. Arg is a multi-valued function.

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

ln(z) = ln(r) + i(q + 2pk)

The principal value is defined by setting k = 0. The principal value has imaginary part in the range (-π,π] and equals the natural logarithm for the positive reals.

The principal value of the logarithm of a negative number is:

ln(-r) = ln(r) + ip

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).

Group theory

From the pure mathematical perspective, the identity

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.

Bases

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.

• 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

• binary logarithm (log₂; sometimes lg, lb, or ld), in computer science and information theory

• 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.

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

Other notations

The notation "ln(x)" invariably means log_e(x), i.e., the natural logarithm of x, but the implied base for "log(x)" varies by discipline:

• Mathematicians understand "log(x)" to mean log_e(x). Calculus textbooks will occasionally write "lg(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, Fortran, 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).

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.

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. 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 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:

• 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.

Change of base

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:

log_b(x) = {log_k(x)}/ {log_k(b)}

Moreover, this result implies that all logarithm functions (whatever the base) are similar to each other. So to calculate the log with base 2 of the number 16 with a calculator:

log₂(16) = {log(16)}/ {log(2)}

Uses of logarithms

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.

Science

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.

• 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 does 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).

Exponential functions

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

 

The antilogarithm function is another name for the inverse of the logarithmic function. It is written antilog_b(n) and means the same as bⁿ.

Generalizations

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, 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.