Editing Napierian logarithm

{{Short description|Mathematical function}}
[[File:NapLog.png|thumb|300px|A plot of the Napierian logarithm for inputs between 0 and 10<sup>8</sup>.]]
[[File:Napier's Mirici Logarithmorum table for 19 deg.agr.jpg|thumb|300px|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.<ref>{{cite book |last1=Larson |first1=Ron |last2=Hostetler |first2=Robert P. |last3=Edwards |first3=Bruce H. |year=2008 |title=Essential Calculus Early Transcendental Functions |publisher=Richard Stratton |isbn=978-0-618-87918-2 |location=U.S.A |pages=119}}</ref><ref name=Hobson>{{Citation|author=Ernest William Hobson|title=John Napier and the Invention of Logarithms, 1614|year=1914|publisher=The University Press|location=Cambridge|url=https://jscholarship.library.jhu.edu/bitstream/handle/1774.2/34187/31151005337641.pdf}}</ref> 
However, if it is taken to mean the "[[logarithm]]s" as originally produced by Napier, it is a function given by (in terms of the modern [[natural logarithm]]): 
: <math>\mathrm{NapLog}(x) = -10^7 \ln (x/10^7) </math>

The Napierian logarithm satisfies identities quite similar to the modern logarithm, such as<ref>{{cite web|last1=Roegel|first1=Denis|title=Napier's ideal construction of the logarithms|url=https://hal.inria.fr/inria-00543934/document|website=HAL|publisher=INRIA|accessdate=7 May 2018}}</ref>
: <math>\mathrm{NapLog}(xy) \approx \mathrm{NapLog}(x)+\mathrm{NapLog}(y)-161180956</math>
or
:<math>\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>\mathrm{NapLog}(\theta) = -10^7 \ln (\sin(\theta)) </math>

== Properties ==

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

: <math>\mathrm{NapLog} (x) \approx 10000000 (16.11809565 - \ln x)</math>

and to the [[common logarithm]] by

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

Note that

: <math>16.11809565 \approx 7 \ln \left(10\right) </math>

and

: <math>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>\ln(.5000000) = -0.6931471806</math>
:<math>\ln(.3333333) = -1.0986123887</math>

would have the corresponding Napierian logarithms:

:<math>\mathrm{NapLog}(5000000) = 6931472</math>
:<math>\mathrm{NapLog}(3333333) = 10986124</math>

For further detail, see [[history of logarithms]].

==References==
{{Reflist}}
*{{citation
 | last1 = Boyer
 | first1 = Carl B.
 | last2 = Merzbach
 | first2 = Uta C.
 | author2-link = Uta Merzbach
 | isbn = 978-0-471-54397-8
 | page = [https://archive.org/details/historyofmathema00boye/page/313 313]
 | publisher = Wiley
 | title = A History of Mathematics
 | year = 1991
 | url-access = registration
 | url = https://archive.org/details/historyofmathema00boye/page/313
 }}.
*{{cite book|author=C.H.Jr. Edwards|title=The Historical Development of the Calculus|url=https://books.google.com/books?id=ilrlBwAAQBAJ&q=napier+logarithm|date=6 December 2012|publisher=Springer Science & Business Media|isbn=978-1-4612-6230-5}}.
*{{citation
 | last = Phillips
 | first = George McArtney
 | isbn = 978-0-387-95022-8
 | page = [https://archive.org/details/twomillenniaofma0000phil/page/61 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) [http://locomat.loria.fr/napier/napier1619construction.pdf Napier’s Ideal Construction of the Logarithms], from the Loria Collection of Mathematical Tables.

[[Category:Logarithms]]