Napierian Logarithm

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):
: $\mathrm(x) = -10^7 \ln (x/10^7)$

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

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

: $\mathrm(\theta) = -10^7 \ln (\sin(\theta))$

Properties

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

: $\mathrm (x) \approx 10000000 (16.11809565 - \ln x)$

and to the common logarithm by

: $\mathrm (x) \approx 23025851 (7 - \log_ x).$

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:

:$\mathrm(5000000) = 6931472$
:$\mathrm(3333333) = 10986124$

For further detail, see history of logarithms.

References

*.
*.
*{{citation
, last = Phillips
, first = George McArtney
, isbn = 978-0-387-95022-8
, page
61
, publisher = Springer-Verlag
, series = CMS Books in Mathematics
, title = Two Millennia of Mathematics: from Archimedes to Gauss
, volume = 6
, year = 2000
, url = 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.

Logarithms