Cologarithm: Difference between revisions

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	:{{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"/>}}		:{{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"/>}}

			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>}}

	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>.

Revision as of 15:24, 6 October 2018

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.

\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:

\operatorname {colog} _{b}(x)=\log _{\frac {1}{b}}\left(x\right)

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 pK_a was later defined as –log₁₀ K_a.

References

^ 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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