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: <math>\operatorname{colog}_b(x) = \log_{\frac{1}{b}}(x).</math> : <math>\operatorname{colog}_b(x) = \log_{\frac{1}{b}}(x).</math>


In ], a decimal cologarithm is indicated by the letter p. This usage originated with the quantity ], defined as −log<sub>10</sub> . Based on pH, the quantity ] was later defined as −log<sub>10</sub> ''K''<sub>a</sub>. In ], a ] cologarithm is indicated by the letter p. This usage originated with the quantity ], defined as −log<sub>10</sub> . Based on pH, the quantity ] was later defined as −log<sub>10</sub> ''K''<sub>a</sub>.


== See also == == See also ==
* ] * ]
* ]


== References == == References ==

Revision as of 13:02, 9 September 2019

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 (January 1909). "Chapter IV. Logarithms Cologarithms". Written at Ann Arbor, Michigan, USA. Trigonometry. Vol. Part I: Plane Trigonometry. New York, USA: Henry Holt and Company / Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA. p. 36. Retrieved 2017-08-12.

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