Napierian Logarithm

The first definition of the logarithm was constructed by Napier and popularized through his posthumous pamphlet (Napier 1619). It this pamphlet, Napier sought to reduce the operations of multiplication, division, and root extraction to addition and subtraction. To this end, he defined the "logarithm"  of a number  by

 

(1)

written .

This definition leads to the remarkable relations

			(2)
			(3)
			(4)

which give the identities

			(5)
			(6)
			(7)

(Havil 2003, pp. 8-9). While Napier's definition is different from the modern one (in particular, it decreases with increasing , but also fails to satisfy a number of properties of the modern logarithm), it provides the desired property of transforming multiplication into addition.

The Napierian logarithm can be given in terms of the modern logarithm by solving equation (1) for , giving

(8)

Because a ratio of logarithms appears in this expression, any logarithm base  can be used as long as the same value of  is used for both numerator and denominator.

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REFERENCES:

Boyer, C. B. and Merzbach, U. C. "Invention of Logarithms." A History of Mathematics, 2nd ed. New York: Wiley, pp. 312-313, 1991.

Gridgeman, N. T. "John Napier and the History of Logarithms." Scripta Math. 29, 49-65, 1969.

Havil, J. "The Baron's Wonderful Canon." §1.2 in Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 4-11, 2003.

Napier, J. The Construction of the Wonderful Canon of Logarithms. 1619. Republished by Blackwood and Sons, 1898.

Napier, J. The Description of Logarithms. 1614.

مواضيع ذات صلة

مشتق الدوال الثابتة : Differntiation Of Constant Functions

اشتقاق جمع وطرح الدوال : Differentiation of Sums and Differences

معادلة خط المستقيم المماس للمنحنى : Equations Of Tangent Lines

ضبط رموز المشتقات : Notations For Derivatives

تعريف مشتق الدالة عند نقطة : DEFINTION OF DERIVATIVE OF FUNCTION IN POINT

الاشتقاق والتفاضل على (IR) DIFFERENTIATION AND DERIVATIVES IN IR

معادلة خط المستقيم المماس للمنحني : TANGENT LINES

نظرية القيم الوسطى : THE INTERMEDIATE – VALUE THEOREN

القيم العظمى والصغرى : EXTREMA – MAXIMA AND MINIMA

دوال كثيرات الحدود والدوال الكسرية : POLYNOMIALS AND RATIONAL FUNCTIONS

الاستمرارية: continuity

اتصال الدوال والاستمرارية CONTINUITY OF FUNCTIONS