Editing Natural logarithm (section)

==History==
{{Main|History of logarithms}}

The concept of the natural logarithm was worked out by [[Gregoire de Saint-Vincent]] and [[Alphonse Antonio de Sarasa]] before 1649.<ref>{{cite journal |author-first=R.P. |author-last=Burn |date=2001 |title=Alphonse Antonio de Sarasa and logarithms |journal=[[Historia Mathematica]] |volume=28|pages=1–17|doi=10.1006/hmat.2000.2295}}</ref> Their work involved [[quadrature (geometry)|quadrature]] of the [[hyperbola]] with equation {{math|1=''xy'' = 1}}, by determination of the area of [[hyperbolic sector]]s. Their solution generated the requisite "[[hyperbolic logarithm]]" [[function (mathematics)|function]], which had the properties now associated with the natural logarithm.

An early mention of the natural logarithm was by [[Nicholas Mercator]] in his work ''Logarithmotechnia'', published in 1668,<ref>{{cite web |author-first1=J. J. |author-last1=O'Connor |author-first2=E. F. |author-last2=Robertson |url=http://www-history.mcs.st-and.ac.uk/HistTopics/e.html |title=The number e |publisher=The MacTutor History of Mathematics archive |date=September 2001 |access-date=2009-02-02}}</ref> although the mathematics teacher [[John Speidell]] had already compiled a table of what in fact were effectively natural logarithms in 1619.<ref name = Cajori>{{cite book |author-last=Cajori |author-first=Florian |author-link=Florian Cajori |title=A History of Mathematics |edition=5th |page=152 |publisher=AMS Bookstore |date=1991 |isbn=0-8218-2102-4 |url=https://books.google.com/books?id=mGJRjIC9fZgC}}</ref> It has been said that Speidell's logarithms were to the base {{mvar|e}}, but this is not entirely true due to complications with the values being expressed as [[integer]]s.<ref name = Cajori/>{{rp|p=152}}