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Gaussian logarithm

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In mathematics, addition and subtraction logarithms or Gaussian logarithms can be utilized to find the logarithms of the sum and difference of a pair of values whose logarithms are known, without knowing the values themselves.

Their mathematical foundations trace back to Zecchini Leonelli and Carl Friedrich Gauss in the early 1800s.

The s b ( z ) {\displaystyle s_{b}(z)} and d b ( z ) {\displaystyle d_{b}(z)} functions for b = e {\displaystyle b=e} .

The operations of addition and subtraction can be calculated by the formulas

log b ⁡ ( | X | + | Y | ) = x + s b ( y − x ) , {\displaystyle \log _{b}{\big (}|X|+|Y|{\big )}=x+s_{b}(y-x),}
log b ⁡ ( | | X | − | Y | | ) = x + d b ( y − x ) , {\displaystyle \log _{b}{\big (}{\big |}|X|-|Y|{\big |}{\big )}=x+d_{b}(y-x),}

where x = log b ⁡ | X | {\displaystyle x=\log _{b}|X|} , y = log b ⁡ | Y | {\displaystyle y=\log _{b}|Y|} , the "sum" function is defined by s b ( z ) = log b ⁡ ( 1 + b z ) {\displaystyle s_{b}(z)=\log _{b}(1+b^{z})} , and the "difference" function by d b ( z ) = log b ⁡ | 1 − b z | {\displaystyle d_{b}(z)=\log _{b}|1-b^{z}|} . The functions s b ( z ) {\displaystyle s_{b}(z)} and d b ( z ) {\displaystyle d_{b}(z)} are also known as Gaussian logarithms.

For natural logarithms with b = e {\displaystyle b=e} the following identities with hyperbolic functions exist:

s e ( z ) = ln ⁡ 2 + z 2 + ln ⁡ ( cosh ⁡ z 2 ) . {\displaystyle s_{e}(z)=\ln 2+{\frac {z}{2}}+\ln \left(\cosh {\frac {z}{2}}\right).}
d e ( z ) = ln ⁡ 2 + z 2 + ln ⁡ | sinh ⁡ z 2 | . {\displaystyle d_{e}(z)=\ln 2+{\frac {z}{2}}+\ln \left|\sinh {\frac {z}{2}}\right|.}

This shows that s e {\displaystyle s_{e}} has a Taylor expansion where all but the first term are rational and all odd terms except the linear term are zero.

The simplification of multiplication, division, roots, and powers is counterbalanced by the cost of evaluating these functions for addition and subtraction.

See also

References

  1. ^ "Logarithm: Addition and Subtraction, or Gaussian Logarithms". Encyclopædia Britannica Eleventh Edition.
  2. ^ Leonelli, Zecchini (1803) . Supplément logarithmique. Théorie des logarithmes additionels et diductifs (in French). Bordeaux: Brossier. (NB. 1802/1803 is the year XI. in the French Republican Calendar.)
  3. ^ Leonhardi, Gottfried Wilhelm (1806). LEONELLIs logarithmische Supplemente, als ein Beitrag, Mängel der gewöhnlichen Logarithmentafeln zu ersetzen. Aus dem Französischen nebst einigen Zusätzen von GOTTFRIED WILHELM LEONHARDI, Souslieutenant beim kurfürstlichen sächsischen Feldartilleriecorps (in German). Dresden: Walther'sche Hofbuchhandlung. (NB. An expanded translation of Zecchini Leonelli's Supplément logarithmique. Théorie des logarithmes additionels et diductifs.)
  4. ^ Gauß, Johann Carl Friedrich (1808-02-12). "LEONELLI, Logarithmische Supplemente". Allgemeine Literaturzeitung (in German) (45). Halle-Leipzig: 353–356.
  5. ^ Dunnington, Guy Waldo (2004) . Gray, Jeremy; Dohse, Fritz-Egbert (eds.). Carl Friedrich Gauss - Titan of Science. Spectrum series (revised ed.). Mathematical Association of America (MAA). ISBN 978-0-88385-547-8. ISBN 0-88385-547-X.

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