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Logarithm
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==Logarithm tables, slide rules, and historical applications{{anchor|Antilogarithm}}== [[Image:Logarithms Britannica 1797.png|thumb|The 1797 ''[[Encyclopædia Britannica]]'' explanation of logarithms]] 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."<ref>{{Citation |last1=Bryant |first1=Walter W. |title=A History of Astronomy |url=https://archive.org/stream/ahistoryastrono01bryagoog#page/n72/mode/2up |publisher=Methuen & Co|location=London |year=1907 }}, p. 44</ref> As the function {{math|''f''(''x'') {{=}} {{mvar|b}}<sup>''x''</sup>}} is the inverse function of {{math|1=log<sub>''b''</sub> ''x''}}, it has been called an '''antilogarithm'''.<ref>{{Citation|editor1-last=Abramowitz|editor1-first=Milton|editor1-link=Milton Abramowitz|editor2-last=Stegun|editor2-first=Irene A.|editor2-link=Irene Stegun|title=Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables|publisher=[[Dover Publications]]|location=New York|isbn=978-0-486-61272-0|edition=10th|year=1972|title-link=Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables}}, section 4.7., p. 89</ref> Nowadays, this function is more commonly called an [[exponential function]]. ===Log tables=== A key tool that enabled the practical use of logarithms was the ''[[log table|table of logarithms]]''.<ref>{{Citation | last1=Campbell-Kelly | first1=Martin | title=The history of mathematical tables: from Sumer to spreadsheets|title-link= The History of Mathematical Tables | publisher=[[Oxford University Press]] | series=Oxford scholarship online | isbn=978-0-19-850841-0 | year=2003}}, section 2</ref> The first such table was compiled by [[Henry Briggs (mathematician)|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 logarithm]]s 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 {{math|log<sub>10</sub> ''x''}} for any number {{mvar|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 {{mvar|x}} can be separated into an [[integer part]] and a [[fractional part]], known as the characteristic and [[mantissa (logarithm)|mantissa]]. Tables of logarithms need only include the mantissa, as the characteristic can be easily determined by counting digits from the decimal point.<ref>{{Citation | last1=Spiegel | first1=Murray R. | last2=Moyer | first2=R.E. | title=Schaum's outline of college algebra | publisher=[[McGraw-Hill]] | location=New York | series=Schaum's outline series | isbn=978-0-07-145227-4 | year=2006}}, p. 264</ref> The characteristic of {{math|10 · {{mvar|x}}}} is one plus the characteristic of {{mvar|x}}, and their mantissas are the same. Thus using a three-digit log table, the logarithm of 3542 is approximated by <math display="block">\begin{align} \log_{10}3542 &= \log_{10}(1000 \cdot 3.542) \\ &= 3 + \log_{10}3.542 \\ &\approx 3 + \log_{10}3.54 \end{align}</math> Greater accuracy can be obtained by [[interpolation]]: <math display="block"> \log_{10}3542 \approx{} 3 + \log_{10}3.54 + 0.2 (\log_{10}3.55-\log_{10}3.54) </math> The value of {{math|10<sup>''x''</sup>}} 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 {{Mvar|c}} and ''{{Mvar|d}}'' were routinely calculated as the sum and difference of their logarithms. The product {{Math|''cd''}} or quotient {{Math|''c''/''d''}} came from looking up the antilogarithm of the sum or difference, via the same table: <math display="block"> cd = 10^{\, \log_{10} c} \, 10^{\,\log_{10} d} = 10^{\,\log_{10} c \, + \, \log_{10} d}</math> and <math display="block">\frac c d = c d^{-1} = 10^{\, \log_{10}c \, - \, \log_{10}d}.</math> 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 [[nth root|roots]] are reduced to multiplications or divisions and lookups by <math display="block">c^d = \left(10^{\, \log_{10} c}\right)^d = 10^{\, d \log_{10} c}</math> and <math display="block">\sqrt[d]{c} = c^\frac{1}{d} = 10^{\frac{1}{d} \log_{10} c}.</math> Trigonometric calculations were facilitated by tables that contained the common logarithms of [[trigonometric function]]s. ===Slide rules=== {{main|Slide rule}} 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: [[Image:Slide rule example2 with labels.svg|center|thumb|Schematic depiction of a slide rule. Starting from 2 on the lower scale, add the distance to 3 on the upper scale to reach the product 6. The slide rule works because it is marked such that the distance from 1 to {{mvar|x}} is proportional to the logarithm of {{mvar|x}}.|alt=A slide rule: two rectangles with logarithmically ticked axes, arrangement to add the distance from 1 to 2 to the distance from 1 to 3, indicating the product 6.]] 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.<ref name="ReferenceA">{{Citation|last1=Maor|first1=Eli|title=E: The Story of a Number|publisher=[[Princeton University Press]]|isbn=978-0-691-14134-3|year=2009|at=sections 1, 13}}</ref>
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