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— CH. 1 · INTRODUCTION —

Logarithm

10 min listen · Ch. 1 of 8
8 sections

  • In 1614, John Napier published a book with a grand Latin title: Mirifici Logarithmorum Canonis Descriptio, the Description of the Wonderful Canon of Logarithms. The word he coined, logarithmus, was stitched together from the Greek logos and arithmos. Behind that ornate name sat a simple, radical idea. A logarithm is the exponent to which a fixed base must be raised to produce a given number. Why would navigators, surveyors, and astronomers seize on this so quickly that it spread across seventeenth-century Europe within years? What did it let them do that arithmetic alone could not? And how did a tool for turning multiplication into addition end up describing earthquakes, prime numbers, musical octaves, and the disorder of a box of gas? The answers run from a Belgian Jesuit chasing the area under a curve to Richard Feynman building a bit-by-bit algorithm at Los Alamos.

  • Addition, multiplication, and exponentiation are three of the most fundamental arithmetic operations, and each has an inverse. Subtraction undoes addition. Division undoes multiplication. The logarithm undoes exponentiation. When a base b is raised to a power y to give a value x, the logarithm runs the process backward, taking x and returning y. For a positive base b not equal to 1, the logarithm of a positive number x is the unique exponent that yields x. The whole apparatus rests on one identity that made logarithms historically irresistible. The logarithm of a product equals the sum of the logarithms of its factors, provided the base and both numbers are positive. That single rule turns a hard multiplication into an easy addition. Related identities followed: the logarithm of a ratio is a difference of logarithms, the logarithm of a p-th power is p times the logarithm, and the logarithm of a p-th root divides the logarithm by p. A change-of-base formula lets any logarithm be computed from logarithms in another base, which is why a scientific calculator only needs to know two.

  • Base 10 is called the common or decimal logarithm, and it slots neatly into the decimal number system. Its value is tied to the number of digits in a positive integer: the digit count is the smallest integer strictly bigger than the logarithm. For 5986, the base-10 logarithm is about 3.78, the next integer up is 4, and 5986 has four digits. The natural logarithm takes the constant e as its base, prized in mathematics and physics for its very simple derivative. The binary logarithm uses base 2 and runs through computer science, information theory, music theory, and photography. In information theory, the natural and binary logarithms correspond to nats and bits as fundamental units. In photography, rescaled base-2 logarithms measure exposure values, light levels, exposure times, lens apertures, and film speeds in stops. When the base is clear from context or simply does not matter, it is dropped, and the meaning shifts by field: base 10 in much of science and engineering, base e in mathematics, base 2 in computer science. The International Organization for Standardization even suggests notations for these bases.

  • The natural logarithm did not begin as a logarithm at all. It began as an attempt to find the area under a rectangular hyperbola, a quadrature. Archimedes had managed the parabola in his work The Quadrature of the Parabola in the third century BC, but the hyperbola resisted every effort. Grégoire de Saint-Vincent, a Belgian Jesuit living in Prague, finally published results on it in 1647. The breakthrough was a hidden pattern: the area behaved so that a geometric progression in the input produced an arithmetic progression in the output. That is the signature of a logarithm. A. A. de Sarasa recognized the link to the older tradition of logarithms, giving rise to the term hyperbolic logarithm as a synonym for the natural logarithm. The idea was soon taken up by Christiaan Huygens and James Gregory. Gottfried Wilhelm Leibniz adopted the notation in 1675 and the following year connected it to the integral. The natural logarithm can in fact be defined directly as the area between the x-axis and the curve of one over t, a definition that leans on nothing more than a simple reciprocal.

  • Henry Briggs compiled the first table of logarithms in 1617, just after Napier's invention, and made one decisive change: he used 10 as the base. His first table held the common logarithms of every integer from 1 to 1000, carried to a precision of 14 digits. The genius of the format lay in splitting each common logarithm into two parts, the integer characteristic and the fractional mantissa. Because numbers differing by factors of 10 share a mantissa, tables only needed to list mantissas, while a quick count of digits supplied the characteristic. A worker multiplying two numbers looked up both logarithms, added them, and looked up the antilogarithm of the sum. Pierre-Simon Laplace praised the result as an 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. A second tool needed no table at all. The non-sliding Gunter's rule appeared shortly after Napier, and William Oughtred turned it into the slide rule, a pair of logarithmic scales that move against each other. Sliding one scale physically adds logarithms, so aligning a distance of 2 against a distance of 3 reads off their product, 6. The slide rule stayed essential for engineers and scientists until the 1970s.

  • The analytic heart of the logarithm is its derivative. The natural logarithm has the strikingly simple derivative of one over x, which makes it the unique antiderivative of that reciprocal taking the value 0 at 1. This simplicity is exactly what earned the natural logarithm its name and helps explain the importance of the constant e. The logarithm is also a transcendental function, a number or function that is not algebraic, in the company of pi and e. Computing logarithms by hand uses several routes. A Taylor series approximates the natural logarithm through partial sums that grow steadily more accurate. A faster series built on the inverse hyperbolic tangent converges more quickly, especially when the input is close to 1. For very high precision, the arithmetic-geometric mean takes over. Sasaki and Kanada showed in 1982 that this method was particularly fast for precisions between 400 and 1000 decimal places, using a formula due to Carl Friedrich Gauss. There is also a beautifully physical method. While at Los Alamos National Laboratory working on the Manhattan Project, Richard Feynman devised a bit-processing algorithm resembling long division, later used in the Connection Machine. It builds a product of distinct factors so that the logarithm reduces to simple addition from a table.

  • A logarithmic scale shrinks wide-ranging quantities into a manageable span, and the decibel is the textbook case. It is based on the common logarithm of ratios: ten times the logarithm of a power ratio, or twenty times the logarithm of a voltage ratio. Decibels quantify the attenuation or amplification of signals, the loudness of sounds, the absorbance of light, and the signal-to-noise ratio. Earthquakes are measured the same way, by the common logarithm of the energy released, in the moment magnitude and Richter scales. The numbers are vivid: a 5.0 quake releases 32 times the energy of a 4.0, and a 6.0 releases 1000 times as much. The brightness of stars, their apparent magnitude, is logarithmic too. In chemistry, the letter p marks the negative decimal logarithm, so pH measures the activity of hydronium ions. Neutral water has a hydronium activity of 10 to the minus 7 moles per litre, giving a pH of 7, while vinegar sits near 3. Semilog graphs scale one axis logarithmically, compressing the jump from 1 million to 1 trillion into the same span as 1 to 1 million. On a log-log graph, a power law appears as a straight line whose slope is the exponent.

  • Logarithms turn up wherever a problem splits in half. Binary search, hunting a value in a sorted list, needs on average a number of comparisons proportional to the logarithm of the list's length. Merge sort, which divides a list and merges the sorted halves, runs in time proportional to N times the logarithm of N. Human perception follows suit. Hick's law ties decision time to the logarithm of the number of choices, Fitts's law links movement time to a logarithm of distance over target size, and the Weber-Fechner law relates stimulus to sensation logarithmically. People with little mathematics education even tend to place numbers on a line by their logarithm, so 10 sits as close to 100 as 100 does to 1000. In number theory, the prime number theorem estimates how many primes lie below x using the natural logarithm. Benford's law predicts the leading digits of real-world data, so about 30 percent of values begin with 1 and 18 percent with 2, and auditors hunt deviations from it to flag fraudulent accounting. In music, base-2 logarithms convert frequency ratios into octaves, semitones, and cents, with the note A at 440 Hz, B-flat at 466 Hz, and B at 493 Hz. The concept reaches further still. The discrete logarithm, the multi-valued inverse of exponentiation in finite groups, is believed to be very hard to compute, and that very difficulty underpins the Diffie-Hellman key exchange that secures cryptographic keys over open channels.

Common questions

Who invented logarithms and when were they introduced?

John Napier introduced logarithms in 1614 in a book titled Mirifici Logarithmorum Canonis Descriptio, the Description of the Wonderful Canon of Logarithms. He coined the term logarithmus from the Greek words logos and arithmos.

What is a logarithm in mathematics?

A logarithm of a number is the exponent to which a fixed base must be raised to produce that number. As a single-variable function, the logarithm to base b is the inverse of exponentiation with base b.

What are the three most common logarithm bases?

The three common bases are 10, e, and 2. Base 10 is the common or decimal logarithm used in science and engineering, base e is the natural logarithm widespread in mathematics and physics, and base 2 is the binary logarithm used in computer science, information theory, music theory, and photography.

Why were logarithms important before calculators?

Logarithms turned multiplication and division into addition and subtraction, because the logarithm of a product is the sum of the logarithms of its factors. Using logarithm tables and slide rules, this replaced tedious multi-digit multiplication with table look-ups and simpler addition, greatly aiding astronomy, surveying, and navigation.

Who made the first table of logarithms?

Henry Briggs compiled the first table of logarithms in 1617, just after Napier's invention, using 10 as the base. His first table contained the common logarithms of all integers from 1 to 1000 with a precision of 14 digits.

How are logarithms used in measuring earthquakes and pH?

Earthquake strength is measured by the common logarithm of the energy released, used in the moment magnitude and Richter scales, so a 6.0 quake releases 1000 times the energy of a 4.0. In chemistry, pH is the negative decimal logarithm of hydronium ion activity, so neutral water has a pH of 7 and vinegar about 3.

What is the natural logarithm and why is it called natural?

The natural logarithm uses the constant e as its base and can be defined as the area under the curve of one over t. It is called natural because its derivative is the very simple expression one over x, which also explains the importance of the constant e.

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93 references cited across the entry

  1. 1John Napier and the invention of logarithms, 1614; a lectureErnest William Hobson — Cambridge University Press — 1914
  2. 2Theory of complex functionsRemmert, Reinhold — Springer-Verlag — 1991
  3. 3Basics Of MathematicsS.K. Kate et al. — Technical Publications — 2009
  4. 4Schaum's outline of theory and problems of elements of statistics. I, Descriptive statistics and probabilityStephen Bernstein et al. — McGraw-Hill — 1999
  5. 5BookAlgebra the Easy WayDouglas Downing — Barron's — 2003
  6. 6BookComplexity Theory: Exploring the limits of efficient algorithmsIngo Wegener — Springer-Verlag — 2005
  7. 7BookInformation TheoryJan C.A. van der Lubbe — Cambridge University Press — 1997
  8. 8BookThe Manual of PhotographyElizabeth Allen et al. — Taylor & Francis — 2011
  9. 9BookIntroduction to Applied Mathematics for Environmental ScienceDavid F. Parkhurst — Springer Science & Business Media — 2007
  10. 10BookPrinciples of Mathematical AnalysisWalter Rudin — McGraw-Hill International — 1984
  11. 11BookAlgorithm Design: Foundations, analysis, and internet examplesMichael T. Goodrich et al. — John Wiley & Sons — 2002
  12. 12ReportInternational Organization for Standardization2019
  13. 13BookMathematics: From the birth of numbersJan Gullberg — W.W. Norton & Co — 1997
  14. 14The Chicago Manual of StyleUniversity of Chicago Press — 2003
  15. 15BookI Want to be a Mathematician: An automathographyP. Halmos — Springer-Verlag — 1985
  16. 16BookUniplanar AlgebraI. Stringham — The Berkeley Press — 1893
  17. 17BookIntroduction to Financial TechnologyRoy S. Freedman — Academic Press — 2006
  18. 18Mirifici Logarithmorum Canonis ConstructioJohn Napier et al. — Andrew Hart — 1619
  19. 19John Napier and the invention of logarithms, 1614Ernest William Hobson — The University Press — 1914
  20. 20Jost Bürgi's method for calculating sinesMenso Folkerts et al. — 2016
  21. 21A brief history of logarithmsR. C. Jr. Pierce — January 1977
  22. 22Mathematics and Its HistoryJ. Stillwell — Springer — 2010
  23. 23A History of AstronomyWalter W. Bryant — Methuen & Co. — 1907
  24. 24Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical TablesDover Publications — 1972
  25. 25The history of mathematical tables: from Sumer to spreadsheetsMartin Campbell-Kelly — Oxford University Press — 2003
  26. 26Schaum's outline of college algebraMurray R. Spiegel et al. — McGraw-Hill — 2006
  27. 27E: The Story of a NumberEli Maor — Princeton University Press — 2009
  28. 28Sets, functions, and logic: an introduction to abstract mathematicsKeith Devlin — Chapman & Hall/CRC — 2004
  29. 29Undergraduate analysisSerge Lang — Springer-Verlag — 1997
  30. 30Foundations of Modern AnalysisJean Dieudonné — Academic Press — 1969
  31. 31Calculation of d/dx(Log(b,x))Wolfram Research
  32. 32Calculus: an intuitive and physical approachMorris Kline — Dover Publications — 1998
  33. 33Calculation of Integrate(ln(x))Wolfram Research
  34. 34Differential and integral calculus. Vol. IRichard Courant — John Wiley & Sons — 1988
  35. 35Gamma: Exploring Euler's ConstantJulian Havil — Princeton University Press — 2003
  36. 36Selected papers on number theory and algebraic geometryKatsumi Nomizu — AMS Bookstore — 1996
  37. 37Transcendental number theoryAlan Baker — Cambridge University Press — 1975
  38. 38Elementary functionsJean-Michel Muller — Birkhäuser Boston — 2006
  39. 39Computer ApproximationsHart et al. — John Wiley — 1968
  40. 40Table driven Newton scheme for high precision logarithm generationM. Zhang et al. — 1994
  41. 41Pseudo Division and Pseudo Multiplication ProcessesJ. E. Meggitt — April 1962
  42. 42Pseudo-Division Algorithms for Floating-Point Logarithms and ExponentialsW. Kahan — 20 May 2001
  43. 43Practically fast multiple-precision evaluation of log(x)T. Sasaki et al. — 1982
  44. 44Stacs 99Timm Ahrendt — Springer — 1999
  45. 45Richard Feynman and The Connection MachineDanny Hillis — 15 January 1989
  46. 46Statistics hacksBruce Frey — O'Reilly — 2006
  47. 47Lectures in applied mathematics and informaticsLuigi M. Ricciardi — Manchester University Press — 1990
  48. 48BookPower QualityC. Sankaran — Taylor & Francis — 2001
  49. 49Springer handbook of acousticsGeorge C. Maling — Springer-Verlag — 2007
  50. 50Sound Capture and Processing: Practical ApproachesIvan Jelev Tashev — John Wiley & Sons — 2009
  51. 51Wavelets: a mathematical tool for signal processingC.K. Chui — Society for Industrial and Applied Mathematics — 1997
  52. 52Functions and Change: A Modeling Approach to College AlgebraBruce Crauder et al. — Cengage Learning — 2008
  53. 53Astronomy methods: a physical approach to astronomical observationsHale Bradt — Cambridge University Press — 2004
  54. 54JournalThe origin and the meaning of the little p in pHNørby, Jens — 2000
  55. 55Compendium of Chemical Terminology ("Gold Book")IUPAC — Blackwell Scientific Publications — 1997
  56. 56Newnes engineering mathematics pocket bookJ.O. Bird — Newnes — 2001
  57. 57Encyclopedia of PerceptionE. Bruce Goldstein — Sage — 2009
  58. 58Human Performance: Cognition, Stress, and Individual DifferencesGerald Matthews — Psychology Press — 2000
  59. 59Fundamentals of skillA.T. Welford — Methuen — 1968
  60. 60The information capacity of the human motor system in controlling the amplitude of movementPaul M. Fitts — June 1954
  61. 61Encyclopaedic dictionary of psychological termsJ.C. Banerjee — M.D. Publications — 1994
  62. 62Encyclopedia of cognitive scienceLynn Nadel — John Wiley & Sons — 2005
  63. 63The Development of Numerical Estimation. Evidence for Multiple Representations of Numerical QuantityRobert S. Siegler et al. — 2003
  64. 64Log or Linear? Distinct Intuitions of the Number Scale in Western and Amazonian Indigene CulturesStanislas Dehaene et al. — 2008
  65. 65ProbabilityLeo Breiman — Society for Industrial and Applied Mathematics — 1992
  66. 66The lognormal distributionJ. Aitchison et al. — Cambridge University Press — 1969
  67. 67An introduction to turbulent flowJean Mathieu and Julian Scott — Cambridge University Press — 2000
  68. 68Mathematical statistics with MathematicaColin Rose et al. — Springer-Verlag — 2002
  69. 69Geometry and BilliardsSerge Tabachnikov — American Mathematical Society — 2005
  70. 70The Effective Use of Benford's Law in Detecting Fraud in Accounting DataCindy Durtschi et al. — 2004
  71. 71Complexity theory: exploring the limits of efficient algorithmsIngo Wegener — Springer-Verlag — 2005
  72. 72Algorithmics: the spirit of computingDavid Harel et al. — Addison-Wesley — 2004
  73. 73The Art of Computer ProgrammingDonald Knuth — Addison-Wesley — 1998
  74. 74Plant physiologyHans Mohr et al. — Springer-Verlag — 1995
  75. 75The open workUmberto Eco — Harvard University Press — 1989
  76. 76Elegant Chaos: Algebraically Simple Chaotic FlowsJulien Clinton Sprott — World Scientific — 2010
  77. 77Getting acquainted with fractalsGilbert Helmberg — Walter de Gruyter — 2007
  78. 78Mathematics and musicDavid Wright — AMS Bookstore — 2009
  79. 79Analytic number theory: an introductory courseP.T. Bateman et al. — World Scientific — 2004
  80. 80An introduction to combinatoricsAlan B. Slomson — CRC Press — 1991
  81. 81Elements of Complex AnalysisS. Ganguly — Academic Publishers — 2005
  82. 82Introduction to complex analysisRolf Herman Nevanlinna et al. — AMS Bookstore — 2007
  83. 83Complex analysisTheral Orvis Moore et al. — World Scientific — 1991
  84. 84Lecture notes on complex analysisIvan Francis Wilde — Imperial College Press — 2006
  85. 85Functions of Matrices. Theory and ComputationNicholas Higham — SIAM — 2008
  86. 86Mathematics of Surfaces XIII: 13th IMA International Conference York, UK, September 7–9, 2009 ProceedingsEdwin R. Hancock et al. — Springer — 2009
  87. 87Cryptography: Theory and PracticeDouglas Robert Stinson — CRC Press — 2006
  88. 88Finite fieldsRudolf Lidl et al. — Cambridge University Press — 1997
  89. 89On the Lambert W functionR. Corless et al. — 1996
  90. 90Learning from data: concepts, theory, and methodsVladimir Cherkassky et al. — John Wiley & Sons — 2007
  91. 91General topology. Chapters 5–10Nicolas Bourbaki — Springer-Verlag — 1998
  92. 92Factorization calculus and geometric probabilityR.V. Ambartzumian — Cambridge University Press — 1990
  93. 93Lectures on vanishing theoremsHélène Esnault et al. — Birkhäuser Verlag — 1992