page_title: History of logarithms
section_title: Natural logarithm
hierarchical_section_title: History of logarithms / Natural logarithm
caption_attribution_description: Italiano: R.p. Gregorii a S.to Vincentio ... Opus geometricum posthumum ad mesolabium per rationum proportionalium novas proprietates. Finem operis mors authoris antevertit. - Gandavi&#160;: typis Balduini Manilii, typographi iurati, sub signo albae columbae, 1668. - [20], 297, [i.e. 291, 1] p., [1] c. di tav. calcogr.&#160;: ill.&#160;; fol .
context_page_description: The history of logarithms is the story of a correspondence between multiplication on the positive real numbers and addition on the real number line that was formalized in seventeenth century Europe and was widely used to simplify calculation until the advent of the digital computer. The Napierian logarithms were published first in 1614. Henry Briggs introduced common logarithms, which were easier to use. Tables of logarithms were published in many forms over four centuries. The idea of logarithms was also used to construct the slide rule, which became ubiquitous in science and engineering until the 1970s. A breakthrough generating the natural logarithm was the result of a search for an expression of area against a rectangular hyperbola, and required the assimilation of a new function into standard mathematics.
context_section_description: In 1649, Alphonse Antonio de Sarasa, a former student of Grégoire de Saint-Vincent, related logarithms to the quadrature of the hyperbola, by pointing out that the area A(t) under the hyperbola from x = 1 to x = t satisfies
  The natural logarithm was first described by Nicholas Mercator in his work Logarithmotechnia published in 1668, although the mathematics teacher John Speidell had already in 1619 compiled a table of what were effectively natural logarithms, based on Napier's work.
  Historian Tom Whiteside described the transition to the analytic function as follows:
  By the end of the 17th century we can say that much more than being a calculating device suitably well-tabulated, the logarithm function, very much on the model of the hyperbola-area, had been accepted into mathematics. When, in the 18th century, this geometric basis was discarded in favour of a fully analytical one, no extension or reformulation was necessary – the concept of "hyperbola-area" was transformed painlessly into "natural logarithm".
  Leonard Euler treated a logarithm as an exponent of a certain number called the base of the logarithm. He noted that the number 2.71828, and its reciprocal, provided a point on the hyperbola xy = 1 such that an area of one square unit lies beneath the hyperbola, right of (1,1) and above the asymptote of the hyperbola. He then called the logarithm, with this number as base, the natural logarithm.
  As noted by Howard Eves, "One of the anomalies in the history of mathematics is the fact that logarithms were discovered before exponents were in use." Carl B. Boyer wrote, "Euler was among the first to treat logarithms as exponents, in the manner now so familiar."
caption: Opus geometricum posthumum, 1668
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