Napierian logarithm

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. 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):

[math]\displaystyle{ \mathrm{NapLog}(x) = -10^7 \ln (x/10^7) }[/math]

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

[math]\displaystyle{ \mathrm{NapLog}(xy) \approx \mathrm{NapLog}(x)+\mathrm{NapLog}(y)-161180956 }[/math]

or

[math]\displaystyle{ \mathrm{NapLog}(xy/10^7) = \mathrm{NapLog}(x)+\mathrm{NapLog}(y) }[/math]

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

[math]\displaystyle{ \mathrm{NapLog}(\theta) = -10^7 \ln (\sin(\theta)) }[/math]

Properties

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

[math]\displaystyle{ \mathrm{NapLog} (x) \approx 10000000 (16.11809565 - \ln x) }[/math]

and to the common logarithm by

[math]\displaystyle{ \mathrm{NapLog} (x) \approx 23025851 (7 - \log_{10} x). }[/math]

Note that

[math]\displaystyle{ 16.11809565 \approx 7 \ln \left(10\right) }[/math]

and

[math]\displaystyle{ 23025851 \approx 10^7 \ln (10). }[/math]

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

[math]\displaystyle{ \ln(.5000000) = -0.6931471806 }[/math]

[math]\displaystyle{ \ln(.3333333) = -1.0986123887 }[/math]

would have the corresponding Napierian logarithms:

[math]\displaystyle{ \mathrm{NapLog}(5000000) = 6931472 }[/math]

[math]\displaystyle{ \mathrm{NapLog}(3333333) = 10986124 }[/math]

For further detail, see history of logarithms.

References

• Boyer, Carl B.; Merzbach, Uta C. (1991), A History of Mathematics, Wiley, p. 313, ISBN 978-0-471-54397-8, https://archive.org/details/historyofmathema00boye/page/313 .
• C.H.Jr. Edwards (6 December 2012). The Historical Development of the Calculus. Springer Science & Business Media. ISBN 978-1-4612-6230-5. https://books.google.com/books?id=ilrlBwAAQBAJ&q=napier+logarithm. .
• Phillips, George McArtney (2000), Two Millennia of Mathematics: from Archimedes to Gauss, CMS Books in Mathematics, 6, Springer-Verlag, p. 61, ISBN 978-0-387-95022-8, https://archive.org/details/twomillenniaofma0000phil/page/61 .

External links

• Denis Roegel (2012) Napier’s Ideal Construction of the Logarithms, from the Loria Collection of Mathematical Tables.