# Napierian logarithm

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. Napier did not introduce this natural logarithmic function, although it is named after him.[1] 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):

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

The Napierian logarithm satisfies identities quite similar to the modern logarithm, such as[2]

${\displaystyle \mathrm {NapLog} (xy)=\mathrm {NapLog} (x)+\mathrm {NapLog} (y)-161180956}$

or

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

## Properties

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

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

and to the common logarithm by

${\displaystyle \mathrm {NapLog} (x)\approx 23025851(7-\log _{10}x).}$

Note that

${\displaystyle 16.11809565\approx 7\ln \left(10\right)}$

and

${\displaystyle 23025851\approx 10^{7}\ln(10).}$

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

${\displaystyle \log(.5000000)=-0.6931471806}$
${\displaystyle \log(.3333333)=-1.0986123887}$

would have the corresponding Napierian logarithms:

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

For further detail, see history of logarithms.

## References

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. ^ Roegel, Denis. "Napier's ideal construction of the logarithms". HAL. INRIA. Retrieved 7 May 2018.
• 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.