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Napierian logarithm

Mathematical function

Napierian logarithm

Mathematical function

A plot of the Napierian logarithm for inputs between 0 and 10<sup>8</sup>.

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. 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): : \mathrm{NapLog}(x) = -10^7 \ln (x/10^7)

The Napierian logarithm satisfies identities quite similar to the modern logarithm, such as : \mathrm{NapLog}(xy) \approx \mathrm{NapLog}(x)+\mathrm{NapLog}(y)-161180956 or :\mathrm{NapLog}(xy/10^7) = \mathrm{NapLog}(x)+\mathrm{NapLog}(y)

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

: \mathrm{NapLog}(\theta) = -10^7 \ln (\sin(\theta))

Properties

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

: \mathrm{NapLog} (x) \approx 10000000 (16.11809565 - \ln x)

and to the common logarithm by

: \mathrm{NapLog} (x) \approx 23025851 (7 - \log_{10} x).

Note that

: 16.11809565 \approx 7 \ln \left(10\right)

and

: 23025851 \approx 10^7 \ln (10).

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

:\ln(.5000000) = -0.6931471806 :\ln(.3333333) = -1.0986123887

would have the corresponding Napierian logarithms:

:\mathrm{NapLog}(5000000) = 6931472 :\mathrm{NapLog}(3333333) = 10986124

For further detail, see history of logarithms.

References

  • {{citation | author2-link = Uta Merzbach | url-access = registration
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References

  1. (2008). "Essential Calculus Early Transcendental Functions". Richard Stratton.
  2. Ernest William Hobson. (1914). "John Napier and the Invention of Logarithms, 1614". The University Press.
  3. "Napier's ideal construction of the logarithms". INRIA.
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