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.

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.