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Common logarithm

In mathematics, the common logarithm is the logarithm with base 10.[1] It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered its use, as well as standard logarithm. Historically, it was known as logarithmus decimalis[2] or logarithmus decadis.[3] It is indicated by log(x),[4] log10(x),[5] or sometimes Log(x) with a capital L;[note 1] on calculators, it is printed as "log", but mathematicians usually mean natural logarithm (logarithm with base e ≈ 2.71828) rather than common logarithm when writing "log". To mitigate this ambiguity, the ISO 80000 specification recommends that log10(x) should be written lg(x), and loge(x) should be ln(x).

A graph of the common logarithm of numbers from 0.1 to 100
Page from a table of common logarithms. This page shows the logarithms for numbers from 1000 to 1509 to five decimal places. The complete table covers values up to 9999.

Before the early 1970s, handheld electronic calculators were not available, and mechanical calculators capable of multiplication were bulky, expensive and not widely available. Instead, tables of base-10 logarithms were used in science, engineering and navigation—when calculations required greater accuracy than could be achieved with a slide rule. By turning multiplication and division to addition and subtraction, use of logarithms avoided laborious and error-prone paper-and-pencil multiplications and divisions.[1] Because logarithms were so useful, tables of base-10 logarithms were given in appendices of many textbooks. Mathematical and navigation handbooks included tables of the logarithms of trigonometric functions as well.[6] For the history of such tables, see log table.

Mantissa and characteristic edit

An important property of base-10 logarithms, which makes them so useful in calculations, is that the logarithm of numbers greater than 1 that differ by a factor of a power of 10 all have the same fractional part. The fractional part is known as the mantissa.[note 2] Thus, log tables need only show the fractional part. Tables of common logarithms typically listed the mantissa, to four or five decimal places or more, of each number in a range, e.g. 1000 to 9999.

The integer part, called the characteristic, can be computed by simply counting how many places the decimal point must be moved, so that it is just to the right of the first significant digit. For example, the logarithm of 120 is given by the following calculation:

 

The last number (0.07918)—the fractional part or the mantissa of the common logarithm of 120—can be found in the table shown. The location of the decimal point in 120 tells us that the integer part of the common logarithm of 120, the characteristic, is 2.

Negative logarithms edit

Positive numbers less than 1 have negative logarithms. For example,

 

To avoid the need for separate tables to convert positive and negative logarithms back to their original numbers, one can express a negative logarithm as a negative integer characteristic plus a positive mantissa. To facilitate this, a special notation, called bar notation, is used:

 

The bar over the characteristic indicates that it is negative, while the mantissa remains positive. When reading a number in bar notation out loud, the symbol   is read as "bar n", so that   is read as "bar 2 point 07918...". An alternative convention is to express the logarithm modulo 10, in which case

 

with the actual value of the result of a calculation determined by knowledge of the reasonable range of the result.[note 3]

The following example uses the bar notation to calculate 0.012 × 0.85 = 0.0102:

 

* This step makes the mantissa between 0 and 1, so that its antilog (10mantissa) can be looked up.

The following table shows how the same mantissa can be used for a range of numbers differing by powers of ten:

Common logarithm, characteristic, and mantissa of powers of 10 times a number
Number Logarithm Characteristic Mantissa Combined form
n = 5 × 10i log10(n) i = floor(log10(n)) log10(n) − i
5 000 000 6.698 970... 6 0.698 970... 6.698 970...
50 1.698 970... 1 0.698 970... 1.698 970...
5 0.698 970... 0 0.698 970... 0.698 970...
0.5 −0.301 029... −1 0.698 970... 1.698 970...
0.000 005 −5.301 029... −6 0.698 970... 6.698 970...

Note that the mantissa is common to all of the 5  ×  10i. This holds for any positive real number   because

 

Since i is a constant, the mantissa comes from  , which is constant for given  . This allows a table of logarithms to include only one entry for each mantissa. In the example of 5  ×  10i, 0.698 970 (004 336 018 ...) will be listed once indexed by 5 (or 0.5, or 500, etc.).

 
Numbers are placed on slide rule scales at distances proportional to the differences between their logarithms. By mechanically adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale, one can quickly determine that 2  ×  3 = 6.

History edit

Common logarithms are sometimes also called "Briggsian logarithms" after Henry Briggs, a 17th century British mathematician. In 1616 and 1617, Briggs visited John Napier at Edinburgh, the inventor of what are now called natural (base-e) logarithms, in order to suggest a change to Napier's logarithms. During these conferences, the alteration proposed by Briggs was agreed upon; and after his return from his second visit, he published the first chiliad of his logarithms.

Because base-10 logarithms were most useful for computations, engineers generally simply wrote "log(x)" when they meant log10(x). Mathematicians, on the other hand, wrote "log(x)" when they meant loge(x) for the natural logarithm. Today, both notations are found. Since hand-held electronic calculators are designed by engineers rather than mathematicians, it became customary that they follow engineers' notation. So the notation, according to which one writes "ln(x)" when the natural logarithm is intended, may have been further popularized by the very invention that made the use of "common logarithms" far less common, electronic calculators.

Numeric value edit

 
The logarithm keys (log for base-10 and ln for base-e) on a typical scientific calculator. The advent of hand-held calculators largely eliminated the use of common logarithms as an aid to computation.

The numerical value for logarithm to the base 10 can be calculated with the following identities:[5]

  or   or  

using logarithms of any available base  

as procedures exist for determining the numerical value for logarithm base e (see Natural logarithm § Efficient computation) and logarithm base 2 (see Algorithms for computing binary logarithms).

Derivative edit

The derivative of a logarithm with a base b is such that

 , so  .[7]

See also edit

Notes edit

  1. ^ The notation Log is ambiguous, as this can also mean the complex natural logarithmic multi-valued function.
  2. ^ This use of the word mantissa stems from an older, non-numerical, meaning: a minor addition or supplement, e.g., to a text. Nowadays, the word mantissa is generally used to describe the fractional part of a floating-point number on computers, though the recommended term is significand.
  3. ^ For example, Bessel, F. W. (1825). "Über die Berechnung der geographischen Längen und Breiten aus geodätischen Vermessungen". Astronomische Nachrichten. 331 (8): 852–861. arXiv:0908.1823. Bibcode:1825AN......4..241B. doi:10.1002/asna.18260041601. S2CID 118630614. gives (beginning of section 8)  ,  . From the context, it is understood that  , the minor radius of the earth ellipsoid in toise (a large number), whereas  , the eccentricity of the earth ellipsoid (a small number).

References edit

  1. ^ a b Hall, Arthur Graham; Frink, Fred Goodrich (1909). "Chapter IV. Logarithms [23] Common logarithms". Trigonometry. Vol. Part I: Plane Trigonometry. New York: Henry Holt and Company. p. 31.
  2. ^ Euler, Leonhard; Speiser, Andreas; du Pasquier, Louis Gustave; Brandt, Heinrich; Trost, Ernst (1945) [1748]. Speiser, Andreas (ed.). Introductio in Analysin Infinitorum (Part 2). 1 (in Latin). Vol. 9. B.G. Teubner. {{cite book}}: |work= ignored (help)
  3. ^ Scherffer, P. Carolo (1772). Institutionum Analyticarum Pars Secunda de Calculo Infinitesimali Liber Secundus de Calculo Integrali (in Latin). Vol. 2. Joannis Thomæ Nob. De Trattnern. p. 198.
  4. ^ "Introduction to Logarithms". www.mathsisfun.com. Retrieved 2020-08-29.
  5. ^ a b Weisstein, Eric W. "Common Logarithm". mathworld.wolfram.com. Retrieved 2020-08-29.
  6. ^ Hedrick, Earle Raymond (1913). Logarithmic and Trigonometric Tables. New York, USA: Macmillan.
  7. ^ "Derivatives of Logarithmic Functions". Math24. 2021-04-14. from the original on 2020-10-01.

Bibliography edit

common, logarithm, this, article, needs, additional, citations, verification, please, help, improve, this, article, adding, citations, reliable, sources, unsourced, material, challenged, removed, find, sources, news, newspapers, books, scholar, jstor, august, . This article needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources Common logarithm news newspapers books scholar JSTOR August 2020 Learn how and when to remove this template message In mathematics the common logarithm is the logarithm with base 10 1 It is also known as the decadic logarithm and as the decimal logarithm named after its base or Briggsian logarithm after Henry Briggs an English mathematician who pioneered its use as well as standard logarithm Historically it was known as logarithmus decimalis 2 or logarithmus decadis 3 It is indicated by log x 4 log10 x 5 or sometimes Log x with a capital L note 1 on calculators it is printed as log but mathematicians usually mean natural logarithm logarithm with base e 2 71828 rather than common logarithm when writing log To mitigate this ambiguity the ISO 80000 specification recommends that log10 x should be written lg x and loge x should be ln x A graph of the common logarithm of numbers from 0 1 to 100 Page from a table of common logarithms This page shows the logarithms for numbers from 1000 to 1509 to five decimal places The complete table covers values up to 9999 Before the early 1970s handheld electronic calculators were not available and mechanical calculators capable of multiplication were bulky expensive and not widely available Instead tables of base 10 logarithms were used in science engineering and navigation when calculations required greater accuracy than could be achieved with a slide rule By turning multiplication and division to addition and subtraction use of logarithms avoided laborious and error prone paper and pencil multiplications and divisions 1 Because logarithms were so useful tables of base 10 logarithms were given in appendices of many textbooks Mathematical and navigation handbooks included tables of the logarithms of trigonometric functions as well 6 For the history of such tables see log table Contents 1 Mantissa and characteristic 1 1 Negative logarithms 2 History 3 Numeric value 4 Derivative 5 See also 6 Notes 7 References 8 BibliographyMantissa and characteristic editAn important property of base 10 logarithms which makes them so useful in calculations is that the logarithm of numbers greater than 1 that differ by a factor of a power of 10 all have the same fractional part The fractional part is known as the mantissa note 2 Thus log tables need only show the fractional part Tables of common logarithms typically listed the mantissa to four or five decimal places or more of each number in a range e g 1000 to 9999 The integer part called the characteristic can be computed by simply counting how many places the decimal point must be moved so that it is just to the right of the first significant digit For example the logarithm of 120 is given by the following calculation log10 120 log10 102 1 2 2 log10 1 2 2 0 07918 displaystyle log 10 120 log 10 left 10 2 times 1 2 right 2 log 10 1 2 approx 2 0 07918 nbsp The last number 0 07918 the fractional part or the mantissa of the common logarithm of 120 can be found in the table shown The location of the decimal point in 120 tells us that the integer part of the common logarithm of 120 the characteristic is 2 Negative logarithms edit Positive numbers less than 1 have negative logarithms For example log10 0 012 log10 10 2 1 2 2 log10 1 2 2 0 07918 1 92082 displaystyle log 10 0 012 log 10 left 10 2 times 1 2 right 2 log 10 1 2 approx 2 0 07918 1 92082 nbsp To avoid the need for separate tables to convert positive and negative logarithms back to their original numbers one can express a negative logarithm as a negative integer characteristic plus a positive mantissa To facilitate this a special notation called bar notation is used log10 0 012 2 0 07918 1 92082 displaystyle log 10 0 012 approx bar 2 0 07918 1 92082 nbsp The bar over the characteristic indicates that it is negative while the mantissa remains positive When reading a number in bar notation out loud the symbol n displaystyle bar n nbsp is read as bar n so that 2 07918 displaystyle bar 2 07918 nbsp is read as bar 2 point 07918 An alternative convention is to express the logarithm modulo 10 in which case log10 0 012 8 07918mod10 displaystyle log 10 0 012 approx 8 07918 bmod 1 0 nbsp with the actual value of the result of a calculation determined by knowledge of the reasonable range of the result note 3 The following example uses the bar notation to calculate 0 012 0 85 0 0102 As found above log10 0 012 2 07918Sincelog10 0 85 log10 10 1 8 5 1 log10 8 5 1 0 92942 1 92942log10 0 012 0 85 log10 0 012 log10 0 85 2 07918 1 92942 2 0 07918 1 0 92942 2 1 0 07918 0 92942 3 1 00860 2 0 00860 log10 10 2 log10 1 02 log10 0 01 1 02 log10 0 0102 displaystyle begin array rll text As found above amp log 10 0 012 approx bar 2 07918 text Since log 10 0 85 amp log 10 left 10 1 times 8 5 right 1 log 10 8 5 amp approx 1 0 92942 bar 1 92942 log 10 0 012 times 0 85 amp log 10 0 012 log 10 0 85 amp approx bar 2 07918 bar 1 92942 amp 2 0 07918 1 0 92942 amp 2 1 0 07918 0 92942 amp 3 1 00860 amp 2 0 00860 amp approx log 10 left 10 2 right log 10 1 02 amp log 10 0 01 times 1 02 amp log 10 0 0102 end array nbsp This step makes the mantissa between 0 and 1 so that its antilog 10mantissa can be looked up The following table shows how the same mantissa can be used for a range of numbers differing by powers of ten Common logarithm characteristic and mantissa of powers of 10 times a number Number Logarithm Characteristic Mantissa Combined formn 5 10i log10 n i floor log10 n log10 n i5 000 000 6 698 970 6 0 698 970 6 698 970 50 1 698 970 1 0 698 970 1 698 970 5 0 698 970 0 0 698 970 0 698 970 0 5 0 301 029 1 0 698 970 1 698 970 0 000 005 5 301 029 6 0 698 970 6 698 970 Note that the mantissa is common to all of the 5 10i This holds for any positive real number x displaystyle x nbsp because log10 x 10i log10 x log10 10i log10 x i displaystyle log 10 left x times 10 i right log 10 x log 10 left 10 i right log 10 x i nbsp Since i is a constant the mantissa comes from log10 x displaystyle log 10 x nbsp which is constant for given x displaystyle x nbsp This allows a table of logarithms to include only one entry for each mantissa In the example of 5 10i 0 698 970 004 336 018 will be listed once indexed by 5 or 0 5 or 500 etc nbsp Numbers are placed on slide rule scales at distances proportional to the differences between their logarithms By mechanically adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale one can quickly determine that 2 3 6 History editMain article History of logarithms Common logarithms are sometimes also called Briggsian logarithms after Henry Briggs a 17th century British mathematician In 1616 and 1617 Briggs visited John Napier at Edinburgh the inventor of what are now called natural base e logarithms in order to suggest a change to Napier s logarithms During these conferences the alteration proposed by Briggs was agreed upon and after his return from his second visit he published the first chiliad of his logarithms Because base 10 logarithms were most useful for computations engineers generally simply wrote log x when they meant log10 x Mathematicians on the other hand wrote log x when they meant loge x for the natural logarithm Today both notations are found Since hand held electronic calculators are designed by engineers rather than mathematicians it became customary that they follow engineers notation So the notation according to which one writes ln x when the natural logarithm is intended may have been further popularized by the very invention that made the use of common logarithms far less common electronic calculators Numeric value edit nbsp The logarithm keys log for base 10 and ln for base e on a typical scientific calculator The advent of hand held calculators largely eliminated the use of common logarithms as an aid to computation The numerical value for logarithm to the base 10 can be calculated with the following identities 5 log10 x ln x ln 10 displaystyle log 10 x frac ln x ln 10 quad nbsp or log10 x log2 x log2 10 displaystyle quad log 10 x frac log 2 x log 2 10 quad nbsp or log10 x logB x logB 10 displaystyle quad log 10 x frac log B x log B 10 quad nbsp using logarithms of any available base B displaystyle B nbsp as procedures exist for determining the numerical value for logarithm base e see Natural logarithm Efficient computation and logarithm base 2 see Algorithms for computing binary logarithms Derivative editThe derivative of a logarithm with a base b is such thatddxlogb x 1xln b displaystyle d over dx log b x 1 over x ln b nbsp so ddxlog10 x 1xln 10 displaystyle d over dx log 10 x 1 over x ln 10 nbsp 7 See also editBinary logarithm Cologarithm Decibel Logarithmic scale Napierian logarithm Significand also commonly called mantissa Notes edit The notation Log is ambiguous as this can also mean the complex natural logarithmic multi valued function This use of the word mantissa stems from an older non numerical meaning a minor addition or supplement e g to a text Nowadays the word mantissa is generally used to describe the fractional part of a floating point number on computers though the recommended term is significand For example Bessel F W 1825 Uber die Berechnung der geographischen Langen und Breiten aus geodatischen Vermessungen Astronomische Nachrichten 331 8 852 861 arXiv 0908 1823 Bibcode 1825AN 4 241B doi 10 1002 asna 18260041601 S2CID 118630614 gives beginning of section 8 log b 6 51335464 displaystyle log b 6 51335464 nbsp log e 8 9054355 displaystyle log e 8 9054355 nbsp From the context it is understood that b 106 51335464 displaystyle b 10 6 51335464 nbsp the minor radius of the earth ellipsoid in toise a large number whereas e 108 9054355 10 displaystyle e 10 8 9054355 10 nbsp the eccentricity of the earth ellipsoid a small number References edit a b Hall Arthur Graham Frink Fred Goodrich 1909 Chapter IV Logarithms 23 Common logarithms Trigonometry Vol Part I Plane Trigonometry New York Henry Holt and Company p 31 Euler Leonhard Speiser Andreas du Pasquier Louis Gustave Brandt Heinrich Trost Ernst 1945 1748 Speiser Andreas ed Introductio in Analysin Infinitorum Part 2 1 in Latin Vol 9 B G Teubner a href Template Cite book html title Template Cite book cite book a work ignored help Scherffer P Carolo 1772 Institutionum Analyticarum Pars Secunda de Calculo Infinitesimali Liber Secundus de Calculo Integrali in Latin Vol 2 Joannis Thomae Nob De Trattnern p 198 Introduction to Logarithms www mathsisfun com Retrieved 2020 08 29 a b Weisstein Eric W Common Logarithm mathworld wolfram com Retrieved 2020 08 29 Hedrick Earle Raymond 1913 Logarithmic and Trigonometric Tables New York USA Macmillan Derivatives of Logarithmic Functions Math24 2021 04 14 Archived from the original on 2020 10 01 Bibliography editAbramowitz Milton Stegun Irene Ann eds 1983 June 1964 Handbook of Mathematical Functions with Formulas Graphs and Mathematical Tables Applied Mathematics Series Vol 55 Ninth reprint with additional corrections of tenth original printing with corrections December 1972 first ed Washington D C New York United States Department of Commerce National Bureau of Standards Dover Publications ISBN 978 0 486 61272 0 LCCN 64 60036 MR 0167642 LCCN 65 12253 Moser Michael 2009 Engineering Acoustics An Introduction to Noise Control Springer p 448 ISBN 978 3 540 92722 8 Poliyanin Andrei Dmitrievich Manzhirov Alexander Vladimirovich 2007 2006 11 27 Handbook of mathematics for engineers and scientists CRC Press p 9 ISBN 978 1 58488 502 3 Retrieved from https en wikipedia org w index php title Common logarithm amp oldid 1195246791, wikipedia, wiki, book, books, library,

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