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Napierian logarithm

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A plot of the Napierian logarithm for inputs between 0 and 108.
The 19 degree pages from Napier's 1614 table of logarithms of trigonometric functions Mirifici Logarithmorum Canonis Descriptio

The term Napierian logarithm or Naperian logarithm, named after John Napier, is often used to mean the natural logarithm. Napier did not introduce this natural logarithmic function, although it is named after him.[1][2] However, if it is taken to mean the "logarithms" as originally produced by Napier, it is a function given by (in terms of the modern natural logarithm):

The Napierian logarithm satisfies identities quite similar to the modern logarithm, such as[3]

or

In Napier's 1614 Mirifici Logarithmorum Canonis Descriptio, he provides tables of logarithms of sines for 0 to 90°, where the values given (columns 3 and 5) are

Properties

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Napier's "logarithm" is related to the natural logarithm by the relation

and to the common logarithm by

Note that

and

Napierian logarithms are essentially natural logarithms with decimal points shifted 7 places rightward and with sign reversed. For instance the logarithmic values

would have the corresponding Napierian logarithms:

For further detail, see history of logarithms.

References

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  1. ^ Larson, Ron; Hostetler, Robert P.; Edwards, Bruce H. (2008). Essential Calculus Early Transcendental Functions. U.S.A: Richard Stratton. p. 119. ISBN 978-0-618-87918-2.
  2. ^ Ernest William Hobson (1914), John Napier and the Invention of Logarithms, 1614 (PDF), Cambridge: The University Press
  3. ^ Roegel, Denis. "Napier's ideal construction of the logarithms". HAL. INRIA. Retrieved 7 May 2018.
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