Logarithm Explained

In mathematics, the logarithm is the inverse function to exponentiation. That means that the logarithm of a number  to the base  is the exponent to which must be raised to produce . For example, since, the logarithm base 

10 of is, or . The logarithm of to base  is denoted as, or without parentheses, . When the base is clear from the context or is irrelevant it is sometimes written .

The logarithm base is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number  as its base; its use is widespread in mathematics and physics because of its very simple derivative. The binary logarithm uses base and is frequently used in computer science.

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:$$\backslash log\_b(xy)\; =\; \backslash log\_b\; x\; +\; \backslash log\_b\; y,$$provided that, and are all positive and . 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 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.

The concept of logarithm as the inverse of exponentiation extends to other mathematical structures as well. However, in general settings, the logarithm tends to be a multi-valued function. For example, the complex logarithm is the multi-valued inverse of the complex exponential function. Similarly, the discrete logarithm is the multi-valued inverse of the exponential function in finite groups; it has uses in public-key cryptography.

Motivation

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, the base, is raised to a certain power, the exponent, to give a value ; this is denoted$$b^y=x.$$For example, raising to the power of gives :

2 ^{3} = 8 .

The logarithm of base is the inverse operation, that provides the output from the input . That is,

y = log _{b} x is equivalent to x = b ^{y} if is a positive real number. (If 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 $$\backslash log\_b(xy)=\backslash log\_b\; x\; +\; \backslash log\_b\; y,$$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.

Definition

Given a positive real number such that, the logarithm of a positive real number with respect to base  is the exponent by which must be raised to yield . In other words, the logarithm of to base  is the unique real number  such that

b ^{y} = x .^[1]

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

An equivalent and more succinct definition is that the function is the inverse function to the function

x\mapsto b ^{x} .

Examples

• , since .
• Logarithms can also be negative: $\backslash log\_2\; \backslash !\; \backslash frac\; =\; -1$ since $2^\; =\; \backslash frac\; =\; \backslash frac.$
• is approximately 2.176, which lies between 2 and 3, just as 150 lies between and .
• For any base , and, since and, respectively.

Logarithmic identities

See main article: List of logarithmic identities.

Several important formulas, sometimes called logarithmic identities or logarithmic laws, relate logarithms to one another.^[2]

Product, quotient, power, and root

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 -th power of a number is  times the logarithm of the number itself; the logarithm of a -th root is the logarithm of the number divided by . 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.

	Formula	Example
Product	$\backslash log\_b(x\; y)\; =\; \backslash log\_b\; x\; +\; \backslash log\_b\; y$	$\backslash log\_3\; 243\; =\; \backslash log\_3\; (9\; \backslash cdot\; 27)\; =\; \backslash log\_3\; 9\; +\; \backslash log\_3\; 27\; =\; 2\; +\; 3\; =\; 5$
Quotient	$\backslash log\_b\; \backslash !\backslash frac\; =\; \backslash log\_b\; x\; -\; \backslash log\_b\; y$	$\backslash log\_2\; 16\; =\; \backslash log\_2\; \backslash !\backslash frac\; =\; \backslash log\_2\; 64\; -\; \backslash log\_2\; 4\; =\; 6\; -\; 2\; =\; 4$
Power	$\backslash log\_b\backslash left(x^p\backslash right)\; =\; p\; \backslash log\_b\; x$	$\backslash log\_2\; 64\; =\; \backslash log\_2\; \backslash left(2^6\backslash right)\; =\; 6\; \backslash log\_2\; 2\; =\; 6$
Root	$\backslash log\_b\; \backslash sqrt[p]\; =\; \backslash frac$	$\backslash log\_\; \backslash sqrt\; =\; \backslash frac\backslash log\_\; 1000\; =\; \backslash frac\; =\; 1.5$

Change of base

The logarithm can be computed from the logarithms of and with respect to an arbitrary base  using the following formula:$$\backslash log\_b\; x\; =\; \backslash frac.$$

Typical scientific calculators calculate the logarithms to bases 10 and .^[3] Logarithms with respect to any base  can be determined using either of these two logarithms by the previous formula:$$\backslash log\_b\; x\; =\; \backslash frac\; =\; \backslash frac.$$

Given a number and its logarithm to an unknown base , the base is given by:

$$b\; =\; x^\backslash frac,$$

which can be seen from taking the defining equation

x = b ^{log _{b} x} = b ^{y} to the power of \tfrac{ 1 }{y}.

Particular bases

Among all choices for the base, three are particularly common. These are, (the irrational mathematical constant and (the binary logarithm). In mathematical analysis, the logarithm base is widespread because of analytical properties explained below. On the other hand, logarithms (the common logarithm) are easy to use for manual calculations in the decimal number system:^[4]

$$\backslash \; \backslash log\_(\backslash \; 10\backslash \; x\backslash )\backslash \; =\backslash \; \backslash log\_\; 10\backslash \; +\backslash \; \backslash log\_\; x\backslash \; =\backslash \; 1\backslash \; +\backslash \; \backslash log\_\; x\; ~.$$

Thus, is related to the number of decimal digits of a positive integer : The number of digits is the smallest integer strictly bigger than ^[5] For example, 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.^[6] 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 and in photography rescaled base 2 logarithms are used to measure exposure values, light levels, exposure times, lens apertures, and film speeds in "stops".^[7]

The abbreviation 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 still often means the base ten logarithm.^[8] In mathematics usually means to the natural logarithm (base).^{[9] [10]} In computer science and information theory, 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.^[11]

Base	Name for logb x	ISO notation	Other notations
2	binary logarithm	^[12]	,,,^[13]
	natural logarithm		,
10	common logarithm		,
	logarithm to base

History

See main article: History of logarithms. 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).^[14] 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," derived from the Greek, literally meaning, "ratio-number," from logos "proportion, ratio, word" + arithmos "number".

The common logarithm of a number is the index of that power of ten which equals the number.^[15] 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.^[16] 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 Leibniz in 1675,^[17] and the next year he connected it to the integral $\backslash int\; \backslash frac\; .$

Before Euler developed his modern conception of complex natural logarithms, Roger Cotes had a nearly equivalent result when he showed in 1714 that$$\backslash log(\backslash cos\; \backslash theta\; +\; i\backslash sin\; \backslash theta)\; =\; i\backslash theta.$$

Logarithm tables, slide rules, and historical applications

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

"...[a]n admirable artifice which, by reducing to a few days the labour of many months, doubles the life of the astronomer, and spares him the errors and disgust inseparable from long calculations."^[18]

As the function is the inverse function of, it has been called an antilogarithm.^[19] Nowadays, this function is more commonly called an exponential function.

Log tables

A key tool that enabled the practical use of logarithms was the table of logarithms.^[20] 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 for any number  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 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.^[21] The characteristic of is one plus the characteristic of, and their mantissas are the same. Thus using a three-digit log table, the logarithm of 3542 is approximated by

$$\backslash log\_3542\; =\; \backslash log\_(1000\; \backslash cdot\; 3.542)\; =\; 3\; +\; \backslash log\_3.542\; \backslash approx\; 3\; +\; \backslash log\_3.54$$

Greater accuracy can be obtained by interpolation:

$$\backslash log\_3542\; \backslash approx\; 3\; +\; \backslash log\_3.54\; +\; 0.2\; (\backslash log\_3.55-\backslash log\_3.54)$$

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

Computations

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

$$cd\; =\; 10^\; \backslash ,\; 10^\; =\; 10^$$and$$\backslash frac\; c\; d\; =\; c\; d^\; =\; 10^.$$

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$$c^d\; =\; \backslash left(10^\backslash right)^d\; =\; 10^$$

and$$\backslash sqrt[d]\; =\; c^\backslash frac\; =\; 10^.$$

Trigonometric calculations were facilitated by tables that contained the common logarithms of trigonometric functions.

Slide rules

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.

Analytic properties

A deeper study of logarithms requires the concept of a function. A function is a rule that, given one number, produces another number.^[22] An example is the function producing the -th power of from any real number , where the base  is a fixed number. This function is written as . When is positive and unequal to 1, we show below that is invertible when considered as a function from the reals to the positive reals.

Existence

Let be a positive real number not equal to 1 and let .

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.^[23] Now, is strictly increasing (for), or strictly decreasing (for), is continuous, has domain

\R , and has range \R _{>} . Therefore, is a bijection from \R to \R _{>0} . In other words, for each positive real number, there is exactly one real number such that b ^{x} = y .

We let

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

Characterization by the product formula

The function can also be essentially characterized by the product formula$$\backslash log\_b(xy)\; =\; \backslash log\_b\; x\; +\; \backslash log\_b\; y.$$More precisely, the logarithm to any base is the only increasing function f from the positive reals to the reals satisfying and^[24] $$f(xy)=f(x)+f(y).$$

Graph of the logarithm function

As discussed above, the function is the inverse to the exponential function

x\mapsto b ^{x} . Therefore, their graphs correspond to each other upon exchanging the - and the -coordinates (or upon reflection at the diagonal line), as shown at the right: a point on the graph of yields a point on the graph of the logarithm and vice versa. As a consequence, diverges to infinity (gets bigger than any given number) if grows to infinity, provided that is greater than one. In that case, is an increasing function. For, tends to minus infinity instead. When approaches zero, goes to minus infinity for (plus infinity for, respectively).

Derivative and antiderivative

Analytic properties of functions pass to their inverses. Thus, as is a continuous and differentiable function, so is . Roughly, a continuous function is differentiable if its graph has no sharp "corners". Moreover, as the derivative of evaluates to by the properties of the exponential function, the chain rule implies that the derivative of is given by$$\backslash frac\; \backslash log\_b\; x\; =\; \backslash frac.$$That is, the slope of the tangent touching the graph of the logarithm at the point equals .

The derivative of is ; this implies that is the unique antiderivative of that has the value 0 for . It is this very simple formula that motivated to qualify as "natural" the natural logarithm; this is also one of the main reasons of the importance of the constant .

The derivative with a generalized functional argument is$$\backslash frac\; \backslash ln\; f(x)\; =\; \backslash frac.$$The quotient at the right hand side is called the logarithmic derivative of . Computing

Notes and References

1. , chapter 1
2. All statements in this section can be found in or, for example.
3. , p. 21
4. Book: Downing, Douglas . 2003 . Algebra the Easy Way . chapter 17, p. 275 . Barron's Educational Series . Hauppauge, NY . Barron's . 978-0-7641-1972-9 .
5. Book: Wegener, Ingo . 2005 . Complexity Theory: Exploring the limits of efficient algorithms . . Berlin, DE / New York, NY . 978-3-540-21045-0 . 20.
6. Book: van der Lubbe , Jan C.A. . 1997 . [{{google books |plainurl=y |id=tBuI_6MQTcwC|page=3}} Information Theory ]. Cambridge University Press . 978-0-521-46760-5 . 3 .
7. Book: Elizabeth . Allen . Sophie . Triantaphillidou . 2011 . [{{google books |plainurl=y |id=IfWivY3mIgAC|page=228}} The Manual of Photography ]. Taylor & Francis . 978-0-240-52037-7 . 228 .
8. Book: Parkhurst , David F. . 2007 . [{{google books |plainurl=y |id=h6yq_lOr8Z4C|page=288 }} Introduction to Applied Mathematics for Environmental Science ]. illustrated . Springer Science & Business Media . 978-0-387-34228-3 . 288 .
9. Book: Michael T. . Goodrich . Michael T. Goodrich . Roberto . Tamassia . Roberto Tamassia . 2002 . Algorithm Design: Foundations, analysis, and internet examples . John Wiley & Sons . 23 . One of the interesting and sometimes even surprising aspects of the analysis of data structures and algorithms is the ubiquitous presence of logarithms ... As is the custom in the computing literature, we omit writing the base of the logarithm when .
10. Book: Rudin, Walter . 1984 . Theorem 3.29 . Principles of Mathematical Analysis . 3rd ed., International student . McGraw-Hill International . Auckland, NZ . 978-0-07-085613-4 .
11. Quantities and units . . Part 2: Mathematics . 2019 . / . International Organization for Standardization.

    See also ISO 80000-2 .

12. Book: Gullberg, Jan . 1997 . Mathematics: From the birth of numbers . New York, NY . W.W. Norton & Co . 978-0-393-04002-9 . registration .
13. Perl . Yehoshua . Reingold . Edward M. . December 1977 . Understanding the complexity of interpolation search . . 6 . 6 . 219–222 ; footnote 1 . 10.1016/0020-0190(77)90072-2.
14. The sequel ... Constructio was published posthumously: Ian Bruce has made an annotated translation of both books (2012), available from 17centurymaths.com.
15. William Gardner (1742) Tables of Logarithms
16. Enrique Gonzales-Velasco (2011) Journey through Mathematics – Creative Episodes in its History, §2.4 Hyperbolic logarithms, p. 117, Springer
17. [Florian Cajori]
18. , p. 44
19. , section 4.7., p. 89
20. , section 2
21. , p. 264
22. , or see the references in function
23. , section III.3
24. item (4.3.1)