Napierian logarithm

File:Napier's Mirici Logarithmorum table for 19 deg.agr.jpg]]

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.{{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}}{{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}}

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):

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

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

: $\mathrm{NapLog}(xy) \approx \mathrm{NapLog}(x)+\mathrm{NapLog}(y)-161180956$

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

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

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

Properties

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

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

and to the common logarithm by

: $\mathrm{NapLog} (x) \approx 23025851 (7 - \log_{10} x).$

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:

: $\mathrm{NapLog}(5000000) = 6931472$

: $\mathrm{NapLog}(3333333) = 10986124$

For further detail, see history of logarithms.

References

{{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