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PART 1

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History

Precursors

The Babylonians sometime in 2000–1600 BC may have got invented the quarter square multiplication algorithm to multiply two quantities using only addition, subtraction and a stand of potager. However it could not be used pertaining to division with no additional desk of reciprocals. Large furniture of one fourth squares were used to easily simplify the exact multiplication of enormous numbers by 1817 onwards until this is superseded by using computers. Michael jordan Stifel published Arithmetica integra in Nuremberg in 1544, which contains a table of integers and capabilities of 2 that is considered a beginning version of your logarithmic desk. In the 16th and early on 17th hundreds of years an algorithm called prosthaphaeresis was used to approx . multiplication and division. This kind of used the trigonometric personality

cosα+cosβ=2cos12α+βcos12α-β

or perhaps similar to convert the epreuve to improvements and stand lookups. On the other hand logarithms are more straightforward and require fewer work. It could be shown using complex figures that this is basically the same technique.

From Napier to Euler

John Napier (1550–1617), the inventor of logarithms

The process of logarithms was widely propounded by John Napier in 1614, in a book titled Mirifici Logarithmorum Canonis Descriptio (Description of the Great Rule of Logarithms). Joost Bürgi independently developed logarithms but published half a dozen years following Napier Johannes Kepler, whom used logarithm tables substantially to put together his Ephemeris and therefore dedicated that to Napier remarked:... the accent in calculation led Justus Byrgius [Joost Bürgi] on the way to these very logarithms many years ahead of Napier's system appeared; but... instead of rearing up his child to get the public advantage he abandoned it inside the birth.

... the accent in calculation led Justus Byrgius [Joost Bürgi] on the way to these very logarithms many years just before Napier's program appeared; but... instead of rearing up his child for the public benefit he abandoned it in the birth.

—Johannes Kepler,  Rudolphine Tables (1627)

By repeated subtractions Napier calculated (1 − 10−7)L for L ranging coming from 1 to 100. The end result for L=100 is definitely approximately 0. 99999 = you − 10−5. Napier then simply calculated the items of these amounts with 107(1 − 10−5)L for L from you to 50, and performed similarly with 0. 9998 ≈ (1 − 10−5)20 and 0. 9 ≈ zero. 99520. These kinds of computations, which occupied two decades, allowed him to give, for just about any number N from a few to 12 million, the number L that resolves the formula

Napier first called L an " artificial number", but afterwards introduced the word " logarithm"  to imply a number that indicates a ratio:  λόγος (logos) meaning proportion, andἀριθμός (arithmos) meaning number. In modern explication, the regards to natural logarithms is:

where the very close approximation corresponds to the observation that

The invention was quickly and widely met with acclaim. The works of Bonaventura Cavalieri (Italy),  Edmund Wingate (France), Xue Fengzuo (China), and Johannes Kepler's Chilias logarithmorum (Germany) helped spread the idea further.

The hyperbola y = 1/x (red curve) and the area from x = 1 to 6 (shaded in orange). In 1647 Grégoire de Saint-Vincent related logarithms for the quadrature from the hyperbola, by simply pointing out that the area f(t) under the hyperbola from x = 1 to x = tsatisfies

The natural logarithm was first described by Nicholas Mercator in his work Logarithmotechnia published in 1668, although the math concepts teacher Steve Speidell experienced already in 1619 put together a table on the organic logarithm. About 1730,  Leonhard Euler defined the exponential function and the all-natural logarithm by

Euler likewise showed which the two features are inverse to one another.

The Applications of Logarithms

1 . Psychology

Logarithms take place in several laws describing man perception: Hick's law offers a logarithmic relation between your...