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log in personal tools donate create account log in contents move to sidebar hide top 1 mantissa and characteristic toggle mantissa and characteristic subsection 1 1 negative logarithms 2 history 3 numeric value 4 derivative 5 see also 6 notes 7 references 8 bibliography toggle the table of contents common logarithm 37 languages العربية azərbaycanca български català کوردی чӑвашла deutsch español eesti euskara فارسی suomi français nordfriisk हिन्दी հայերեն bahasa indonesia 日本語 қазақша 한국어 bahasa melayu nederlands occitan polski português română русский සිංහල தமிழ் ไทย tagalog türkçe українська oʻzbekcha ўзбекча tiếng việt 閩南語 bân lâm gí 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikifunctions wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia redirected from mantissa logarithm mathematical function this article needs more citations 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 message a graph of the common logarithm of numbers from 0 1 to 100 in mathematics the common logarithm aka standard logarithm is the logarithm with base 10 1 it is also known as the decadic logarithm the decimal logarithm and the briggsian logarithm the name briggsian logarithm is in honor of the british mathematician henry briggs who conceived of and developed the values for the common logarithm historically the common logarithm was known by its latin name logarithmus decimalis 2 or logarithmus decadis 3 the mathematical notation for using the common logarithm is log x 4 log 10 x 5 or sometimes log x with a capital l a on calculators it is printed as log 6 but mathematicians usually mean natural logarithm logarithm with base e 2 71828 rather than common logarithm when writing log since the natural logarithm is contrary to what the name of the common logarithm implies the most commonly used logarithm in pure math 7 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 8 for the history of such tables see log table 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 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 b 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 log 10 120 log 10 10 2 1 2 2 log 10 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 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 by applying this reasoning it can be seen that log 10 120 2 07918 displaystyle log _ 10 120 2 07918 log 10 12 1 07918 displaystyle log _ 10 12 1 07918 and log 10 1 2 0 07918 displaystyle log _ 10 1 2 0 07918 negative logarithms edit positive numbers less than 1 have negative logarithms for example log 10 0 012 log 10 10 2 1 2 2 log 10 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 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 log 10 0 012 2 0 07918 1 92082 displaystyle log _ 10 0 012 approx bar 2 0 07918 1 92082 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 is read as bar n so that 2 07918 displaystyle bar 2 07918 is read as bar 2 point 07918 an alternative convention is to express the logarithm modulo 10 in which case log 10 0 012 8 07918 mod 1 0 displaystyle log _ 10 0 012 approx 8 07918 bmod 1 0 with the actual value of the result of a calculation determined by knowledge of the reasonable range of the result c the following example uses the bar notation to calculate 0 012 0 85 0 0102 as found above log 10 0 012 2 07918 since log 10 0 85 log 10 10 1 8 5 1 log 10 8 5 1 0 92942 1 92942 log 10 0 012 0 85 log 10 0 012 log 10 0 85 2 07918 1 92942 2 0 07918 1 0 92942 2 1 0 07918 0 92942 3 1 00860 2 0 00860 log 10 10 2 log 10 1 02 log 10 0 01 1 02 log 10 0 0102 displaystyle begin array rll text as found above log _ 10 0 012 approx bar 2 07918 text since log _ 10 0 85 log _ 10 left 10 1 times 8 5 right 1 log _ 10 8 5 approx 1 0 92942 bar 1 92942 log _ 10 0 012 times 0 85 log _ 10 0 012 log _ 10 0 85 approx bar 2 07918 bar 1 92942 2 0 07918 1 0 92942 2 1 0 07918 0 92942 3 1 00860 2 0 00860 approx log _ 10 left 10 2 right log _ 10 1 02 log _ 10 0 01 times 1 02 log _ 10 0 0102 end array this step makes the mantissa between 0 and 1 so that its antilog 10 mantissa 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 10 i log 10 n i floor log 10 n log 10 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 10 i this holds for any positive real number x 10 i displaystyle x times 10 i because log 10 x 10 i log 10 x log 10 10 i log 10 x i displaystyle log _ 10 left x times 10 i right log _ 10 x log _ 10 left 10 i right log _ 10 x i since i is a constant the mantissa comes from log 10 x displaystyle log _ 10 x which is constant for given x displaystyle x this allows a table of logarithms to include only one entry for each mantissa in the example of 5 10 i 0 698 970 004 336 018 will be listed once indexed by 5 or 0 5 or 500 etc history edit main 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 log 10 x mathematicians on the other hand wrote log x when they meant log e 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 to mitigate the ambiguity the iso 80000 specification recommends that log e x should be ln x while log 10 x should be written lg x which unfortunately is used for the base 2 logarithm by clrs and sedgwick and the chicago manual of style 10 11 12 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 log 10 x ln x ln 10 log 2 x log 2 10 log b x log b 10 displaystyle log _ 10 x frac ln x ln 10 frac log _ 2 x log _ 2 10 frac log _ b x log _ b 10 using logarithms of any available base b displaystyle b 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 13 d d x log b x 1 x ln b displaystyle d over dx log _ b x 1 over x ln b therefore d d x log 10 x 1 x ln 10 0 4343 x displaystyle d over dx log _ 10 x 1 over x ln 10 approx 0 4343 over x 4 significant digits see also edit binary logarithm cologarithm decibel logarithmic scale napierian logarithm significand also commonly called mantissa scientific notation orders of magnitude numbers 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 citation needed the word was introduced by henry briggs 9 the word mantissa is often used to describe the part of a floating point number that represents its significant digits although significand was the term used for this by ieee 754 and may be preferred to avoid confusion with logarithm mantissas 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 log b 6 51335464 displaystyle log b 6 51335464 log e 8 9054355 displaystyle log e 8 9054355 from the context it is understood that b 10 6 51335464 displaystyle b 10 6 51335464 the minor radius of the earth ellipsoid in toise a large number whereas e 10 8 9054355 10 displaystyle e 10 8 9054355 10 the eccentricity of the earth ellipsoid a small number references edit 1 2 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 1748 chapter 22 solutio nonnullorum problematum ad circulum pertinentium introductio in analysin infinitorum part 2 in latin lausanne marcum michaelem bousquet p 304 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 introduction to logarithms www mathsisfun com retrieved 2020 08 29 1 2 weisstein eric w common logarithm mathworld wolfram com retrieved 2020 08 29 using a calculator laws of logarithms and exponents higher maths revision bbc bitesize bbc retrieved 2025 07 08 introduction to logarithms www mathsisfun com retrieved 2025 07 08 hedrick earle raymond 1913 logarithmic and trigonometric tables new york usa macmillan schwartzman steven 1994 12 31 the words of mathematics an etymological dictionary of mathematical terms in english american mathematical soc p 131 isbn 978 1 61444 501 2 cormen thomas h leiserson charles e rivest ronald l stein clifford 2001 1990 introduction to algorithms 2nd ed mit press and mcgraw hill pp 34 53 54 isbn 0 262 03293 7 sedgewick robert wayne kevin daniel 2011 algorithms addison wesley professional p 185 isbn 978 0 321 57351 3 the chicago manual of style 25th ed university of chicago press 2003 p 530 derivatives of logarithmic functions math24 2021 04 14 archived from the original on 2020 10 01 bibliography edit abramowitz 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 möser 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 authority 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