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== References == == References ==
{{reflist|refs= {{reflist|refs=
<ref name="Hall_1909">{{cite book |title=Trigonometry |volume=Part I: Plane Trigonometry |author-first1=Arthur Graham |author-last1=Hall |author-first2=Fred Goodrich |author-last2=Frink |date=January 1909 |location=Ann Arbor, Michigan, USA |chapter=Chapter IV. Logarithms Cologarithms |publisher=] / Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA |publication-place=New York, USA |page=36 |url=https://archive.org/stream/planetrigonometr00hallrich#page/n51/mode/1up |access-date=2017-08-12 }}</ref> <ref name="Hall_1909">{{cite book |title=Trigonometry |volume=Part I: Plane Trigonometry |first1=Arthur Graham |last1=Hall |first2=Fred Goodrich |last2=Frink |date=1909 |chapter=Chapter IV. Logarithms Cologarithms |publisher=] |location=New York |page=36 |url=https://archive.org/stream/planetrigonometr00hallrich#page/n51/mode/1up}}</ref>
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Revision as of 13:06, 19 March 2020

In mathematics, the base-b cologarithm, sometimes shortened to colog, of a number is the base-b logarithm of the reciprocal of the number. It is equal to the negative base-b logarithm of the number:

colog b ⁡ ( x ) = log b ⁡ ( 1 x ) = log b ⁡ ( 1 ) − log b ⁡ ( x ) = − log b ⁡ ( x ) . {\displaystyle \operatorname {colog} _{b}(x)=\log _{b}\left({\frac {1}{x}}\right)=\log _{b}(1)-\log _{b}(x)=-\log _{b}(x).}

The cologarithm in base b of a number is also equal to the logarithm of the same number having the reciprocal of b as the base:

colog b ⁡ ( x ) = log 1 b ⁡ ( x ) . {\displaystyle \operatorname {colog} _{b}(x)=\log _{\frac {1}{b}}(x).}

In chemistry, a decimal cologarithm is indicated by the letter p. This usage originated with the quantity pH, defined as −log10 . Based on pH, the quantity pKa was later defined as −log10 Ka.

See also

References

  1. ^ Hall, Arthur Graham; Frink, Fred Goodrich (1909). "Chapter IV. Logarithms Cologarithms". Trigonometry. Vol. Part I: Plane Trigonometry. New York: Henry Holt and Company. p. 36.

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