Napierian logarithm

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A plot of the Napierian logarithm for inputs between 0 and 108.

The term Napierian logarithm or Naperian logarithm, named after John Napier, is often used to mean the natural logarithm. 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 logarithm):

$\mathrm{NapLog}(x) = \frac{\log \frac{10^7}{x}}{\log \frac{10^7}{10^7 - 1}}.$

(Since this is a quotient of logarithms, the base of the logarithm chosen is irrelevant.)

It is not a logarithm to any particular base in the modern sense of the term; however, it can be rewritten as:

$\mathrm{NapLog}(x) = \log_{\frac{10^7}{10^7 - 1}} 10^7 - \log_{\frac{10^7}{10^7 - 1}} x$

and hence it is a linear function of a particular logarithm, and so satisfies identities quite similar to the modern logarithm, such as

$\mathrm{NapLog}(xy) = \mathrm{NapLog}(x)+\mathrm{NapLog}(y)-161180950$

Properties

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

$\mathrm{NapLog} (x) \approx 9999999.5 (16.11809565 - \ln x)$

and to the common logarithm by

$\mathrm{NapLog} (x) \approx 23025850 (7 - \log_{10} x).$

Note that

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

and

$23025850 \approx 10^7 \ln (10).$

For further detail, see Logarithm: from Napier to Euler.

References

• Boyer, Carl B.; Merzbach, Uta C. (1991), A History of Mathematics, Wiley, p. 313, ISBN 978-0-471-54397-8.
• Edwards, Charles Henry (1994), The Historical Development of the Calculus, Springer-Verlag, p. 153.
• 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.