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In ], the base-''b'' '''cologarithm''',<ref name="Hall_1909"/> sometimes shortened to '''colog''',<ref name="Hall_1909"/> of a number is the base-''b'' ] of the ] of the number. It is equal to the ''negative'' base-''b'' logarithm of the number. In ], the base-''b'' '''cologarithm''',<ref name="Hall_1909"/> sometimes shortened to '''colog''',<ref name="Hall_1909"/> of a number is the base-''b'' ] of the ] of the number. It is equal to the ''negative'' base-''b'' logarithm of the number:<ref name="Hall_1909"/>


:{{nowrap|<math> \operatorname{colog}_b (x) = \log_b \left(\frac{1}{x} \right) = \log_b (1) - \log_b (x) = -\log_b (x)</math><ref name="Hall_1909"/>}} : <math>\operatorname{colog}_b(x) = \log_b\left(\frac{1}{x}\right) = \log_b(1) - \log_b(x) = -\log_b(x).</math>


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: 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:


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


== See also == == See also ==

Revision as of 01:28, 5 February 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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