Logarithm

In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the 3rd power: 1000 = 10 = 10 × 10 × 10. More generally, if x = b, then y is the logarithm of x to base b, written log_b x = y, so log₁₀ 1000 = 3. As a single-variable function, the logarithm to base b is the inverse of exponentiation with base b.

The logarithm base 10 is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number e ≈ 2.718 as its base; its use is widespread in mathematics and physics because of its very simple derivative. The binary logarithm uses base 2 and is widely used in computer science, information theory, music theory, and photography. When the base is unambiguous from the context or irrelevant it is often omitted, and the logarithm is written log x.

Logarithms were introduced by John Napier in 1614 as a means of simplifying calculations. They were rapidly adopted by navigators, scientists, engineers, surveyors, and others to perform high-accuracy computations more easily. Using logarithm tables, tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition. This is possible because the logarithm of a product is the sum of the logarithms of the factors:

\[\log_b(xy) = \log_b x + \log_b y,\]

provided that b, x and y are all positive and b ≠ 1. The slide rule, also based on logarithms, allows quick calculations without tables, but at lower precision. The present-day notion of logarithms comes from Leonhard Euler, who connected them to the exponential function in the 18th century, and who also introduced the letter e as the base of natural logarithms.

Logarithmic scales reduce wide-ranging quantities to smaller scopes. For example, the decibel (dB) is a unit used to express ratio as logarithms, mostly for signal power and amplitude (of which sound pressure is a common example). In chemistry, pH is a logarithmic measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and of geometric objects called fractals. They help to describe frequency ratios of musical intervals, appear in formulas counting prime numbers or approximating factorials, inform some models in psychophysics, and can aid in forensic accounting.

Condensed: the full section is in Wikipedia.

Addition, multiplication, and exponentiation are three of the most fundamental arithmetic operations. The inverse of addition is subtraction, and the inverse of multiplication is division. Similarly, a logarithm is the inverse operation of exponentiation. Exponentiation is when a number b, the base, is raised to a certain power y, the exponent, to give a value x; this is denoted

\[b^y=x.\]

For example, raising 2 to the power of 3 gives 8: \(2^3 = 8.\)

The logarithm of base b is the inverse operation, that provides the output y from the input x. That is, \(y = \log_b x\) is equivalent to \(x=b^y\) if b is a positive real number. (If b is not a positive real number, both exponentiation and logarithm can be defined but may take several values, which makes definitions much more complicated.)

One of the main historical motivations of introducing logarithms is the formula

\[\log_b(xy)=\log_b x + \log_b y,\]

by which tables of logarithms allow multiplication and division to be reduced to addition and subtraction, a great aid to calculations before the invention of computers.

Given a positive real number b such that b ≠ 1, the logarithm of a positive real number x with respect to base b is the exponent by which b must be raised to yield x. In other words, the logarithm of x to base b is the unique real number y such that \(b^y = x\).

The logarithm is denoted "log_b x" (pronounced as "the logarithm of x to base b", "the base-b logarithm of x", or most commonly "the log, base b, of x").

An equivalent and more succinct definition is that the function log_b is the inverse function to the function \(x\mapsto b^x\).

• log₂ 16 = 4, since 2 = 2 × 2 × 2 × 2 = 16.
• Logarithms can also be negative: \(\log_2 \! \frac{1}{2} = -1\) since \(2^{-1} = \frac{1}{2^1} = \frac{1}{2}.\)
• log₁₀ 150 is approximately 2.176, which lies between 2 and 3, just as 150 lies between 10 = 100 and 10 = 1000.
• For any base b, log_b b = 1 and log_b 1 = 0, since b = b and b = 1, respectively.

The logarithm of a product is the sum of the logarithms of the numbers being multiplied; the logarithm of the ratio of two numbers is the difference of the logarithms. The logarithm of the p-th power of a number is p times the logarithm of the number itself; the logarithm of a p-th root is the logarithm of the number divided by p. The following table lists these identities with examples. Each of the identities can be derived after substitution of the logarithm definitions \(x = b^{\, \log_b x}\) or \(y = b^{\, \log_b y}\) in the left hand sides. In the following formulas, ⁠\(x\)⁠ and ⁠\(y\)⁠ are positive real numbers and ⁠\(p\)⁠ is an integer greater than 1.

The logarithm log_b x can be computed from the logarithms of x and b with respect to an arbitrary base k using the following formula:

\[\log_b x = \frac{\log_k x}{\log_k b}.\]

Typical scientific calculators calculate the logarithms to bases 10 and e. Logarithms with respect to any base b can be determined using either of these two logarithms by the previous formula:

\[\log_b x = \frac{\log_{10} x}{\log_{10} b} = \frac{\log_{e} x}{\log_{e} b}.\]

Given a number x and its logarithm y = log_b x to an unknown base b, the base is given by:

\[b = x^\frac{1}{y},\]

which can be seen from taking the defining equation \(x = b^{\,\log_b x} = b^y\) to the power of \(\tfrac{1}{y}.\)

Among all choices for the base, three are particularly common. These are b = 10, b = e (the irrational mathematical constant e ≈ 2.71828183 ), and b = 2 (the binary logarithm). In mathematical analysis, the logarithm base e is widespread because of analytical properties explained below. On the other hand, base 10 logarithms (the common logarithm) are easy to use for manual calculations in the decimal number system:

\[\log_{10}\,(\,10\,x\,)\ =\;\log_{10} 10\ +\;\log_{10} x\ =\ 1\,+\,\log_{10} x\,.\]

Thus, log₁₀ (x) is related to the number of decimal digits of a positive integer x: The number of digits is the smallest integer strictly bigger than log₁₀ (x) . For example, log₁₀(5986) is approximately 3.78 . The next integer above it is 4, which is the number of digits of 5986. Both the natural logarithm and the binary logarithm are used in information theory, corresponding to the use of nats or bits as the fundamental units of information, respectively. Binary logarithms are also used in computer science, where the binary system is ubiquitous; in music theory, where a pitch ratio of two (the octave) is ubiquitous and the number of cents between any two pitches is a scaled version of the binary logarithm, or log 2 times 1200, of the pitch ratio (that is, 100 cents per semitone in conventional equal temperament), or equivalently the log base 2  ; and in photography, where rescaled base 2 logarithms are used to measure exposure values, light levels, exposure times, lens apertures, and film speeds in "stops".

The abbreviation log x is often used when the intended base can be inferred based on the context or discipline, or when the base is indeterminate or immaterial. Common logarithms (base 10), historically used in logarithm tables and slide rules, are a basic tool for measurement and computation in many areas of science and engineering; in these contexts log x still often means the base ten logarithm. In mathematics log x usually refers to the natural logarithm (base e). In computer science and information theory, log often refers to binary logarithms (base 2). The following table lists common notations for logarithms to these bases. The "ISO notation" column lists designations suggested by the International Organization for Standardization.

The history of logarithms in seventeenth-century Europe saw the discovery of a new function that extended the realm of analysis beyond the scope of algebraic methods. The method of logarithms was publicly propounded by John Napier in 1614, in a book titled Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Canon of Logarithms). Prior to Napier's invention, there had been other techniques of similar scopes, such as the prosthaphaeresis or the use of tables of progressions, extensively developed by Jost Bürgi around 1600. Napier coined the term for logarithm in Middle Latin, logarithmus, literally meaning 'ratio-number', derived from the Greek logos 'proportion, ratio, word' + arithmos 'number'.

The common logarithm of a number is the index of that power of ten which equals the number. Speaking of a number as requiring so many figures is a rough allusion to common logarithm, and was referred to by Archimedes as the "order of a number". The first real logarithms were heuristic methods to turn multiplication into addition, thus facilitating rapid computation. Some of these methods used tables derived from trigonometric identities. Such methods are called prosthaphaeresis.

Invention of the function now known as the natural logarithm began as an attempt to perform a quadrature of a rectangular hyperbola by Grégoire de Saint-Vincent, a Belgian Jesuit residing in Prague. Archimedes had written The Quadrature of the Parabola in the third century BC, but a quadrature for the hyperbola eluded all efforts until Saint-Vincent published his results in 1647. The relation that the logarithm provides between a geometric progression in its argument and an arithmetic progression of values, prompted A. A. de Sarasa to make the connection of Saint-Vincent's quadrature and the tradition of logarithms in prosthaphaeresis, leading to the term "hyperbolic logarithm", a synonym for natural logarithm. Soon the new function was appreciated by Christiaan Huygens, and James Gregory. The notation Log y was adopted by Gottfried Wilhelm Leibniz in 1675, and the next year he connected it to the integral \(\int \frac{dy}{y} .\)

Before Euler developed his modern conception of complex natural logarithms, Roger Cotes had a nearly equivalent result when he showed in 1714 that

\[\log(\cos \theta + i\sin \theta) = i\theta.\]

By simplifying difficult calculations before calculators and computers became available, logarithms contributed to the advance of science, especially astronomy. They were critical to advances in surveying, celestial navigation, and other domains. Pierre-Simon Laplace called logarithms

As the function f(x) = b is the inverse function of log_b x, it has been called an antilogarithm. Nowadays, this function is more commonly called an exponential function.

A key tool that enabled the practical use of logarithms was the table of logarithms. The first such table was compiled by Henry Briggs in 1617, immediately after Napier's invention but with the innovation of using 10 as the base. Briggs' first table contained the common logarithms of all integers in the range from 1 to 1000, with a precision of 14 digits. Subsequently, tables with increasing scope were written. These tables listed the values of log₁₀ x for any number x in a certain range, at a certain precision. Base-10 logarithms were universally used for computation, hence the name common logarithm, since numbers that differ by factors of 10 have logarithms that differ by integers. The common logarithm of x can be separated into an integer part and a fractional part, known as the characteristic and mantissa. Tables of logarithms need only include the mantissa, as the characteristic can be easily determined by counting digits from the decimal point. The characteristic of 10 · x is one plus the characteristic of x, and their mantissas are the same. Thus using a three-digit log table, the logarithm of 3542 is approximated by

\[\begin{align} \log_{10}3542 &= \log_{10}(1000 \cdot 3.542) \\ &= 3 + \log_{10}3.542 \\ &\approx 3 + \log_{10}3.54 \end{align}\]

Greater accuracy can be obtained by interpolation:

\[\log_{10}3542 \approx{} 3 + \log_{10}3.54 + 0.2 (\log_{10}3.55-\log_{10}3.54)\]

The value of 10 can be determined by reverse look up in the same table, since the logarithm is a monotonic function.

The product and quotient of two positive numbers c and d were routinely calculated as the sum and difference of their logarithms. The product cd or quotient c/d came from looking up the antilogarithm of the sum or difference, via the same table:

\[cd = 10^{\, \log_{10} c} \, 10^{\,\log_{10} d} = 10^{\,\log_{10} c \, + \, \log_{10} d}\]

and

\[\frac c d = c d^{-1} = 10^{\, \log_{10}c \, - \, \log_{10}d}.\]

For manual calculations that demand any appreciable precision, performing the lookups of the two logarithms, calculating their sum or difference, and looking up the antilogarithm is much faster than performing the multiplication by earlier methods such as prosthaphaeresis, which relies on trigonometric identities.

Calculations of powers and roots are reduced to multiplications or divisions and lookups by

\[c^d = \left(10^{\, \log_{10} c}\right)^d = 10^{\, d \log_{10} c}\]

Condensed: the full section is in Wikipedia.

Another critical application was the slide rule, a pair of logarithmically divided scales used for calculation. The non-sliding logarithmic scale, Gunter's rule, was invented shortly after Napier's invention. William Oughtred enhanced it to create the slide rule, a pair of logarithmic scales movable with respect to each other. Numbers are placed on sliding scales at distances proportional to the differences between their logarithms. Sliding the upper scale appropriately amounts to mechanically adding logarithms, as illustrated here:

For example, adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale yields a product of 6, which is read off at the lower part. The slide rule was an essential calculating tool for engineers and scientists until the 1970s, because it allows, at the expense of precision, much faster computation than techniques based on tables.

A deeper study of logarithms requires the concept of a function. A function is a rule that, given one number, produces another number. An example is the function producing the \(x\)-th power of \(b\) from any real number \(x\), where the base \(b\) is a fixed number. This function is written as \(f(x)=b^x\). When \(b\) is positive and unequal to \(1\), we show below that \(f\) is invertible when considered as a function from the reals to the positive reals.

Let b be a positive real number not equal to \(1\) and let \(f(x)=b^x\).

It is a standard result in real analysis that any continuous strictly monotonic function is bijective between its domain and range. This fact follows from the intermediate value theorem. Now, \(f\) is strictly increasing (for \(b > 1\)), or strictly decreasing (for \(0 < b < 1\)), is continuous, has domain \(\R\), and has range \(\R_{> 0}\). Therefore, \(f\) is a bijection from \(\R\) to \(\R_{>0}\). In other words, for each positive real number \(y\), there is exactly one real number \(x\) such that \(b^x = y\).

We let \(\log_b\colon\R_{>0}\to\R\) denote the inverse of \(f\). That is, \(\log_by\) is the unique real number \(x\) such that \(b^x = y\). This function is called the base-\(b\) logarithm function or logarithmic function (or just logarithm).