Cologarithm

In mathematics, the base-b cologarithm,[1] sometimes shortened to colog,[1] 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:[1]

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

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:

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

In chemistry, a decimal cologarithm is indicated by the letter p. This usage originated with the quantity pH, defined as −log10 [H3O+]. 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 [28] Cologarithms". Written at Ann Arbor, Michigan, USA. Trigonometry. 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.