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In ], the base-''b'' '''cologarithm''', sometimes shortened to '''colog''', 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.


:<math> \operatorname{colog}_b\ x = \log_b \left(\frac{1}{x} \right) = \log_b 1-\log_b x = -\log_b x </math> :{{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"/>}}


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>.


== References == == References ==
{{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 |dead-url=no}}</ref>
}}


==Further reading==
* *
* *

Revision as of 15:15, 12 August 2017

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)}

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.

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. {{cite book}}: Unknown parameter |dead-url= ignored (|url-status= suggested) (help)

Further reading

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