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Logarithm
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===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 Β· {{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]].
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