A Numerical Approach to Natural Logarithms

Abstract

Of these three approaches, only the last possesses the virtue of presenting In as a logarithm from the outset. All three approaches are, of course, mathematically valid. The choice between them depends, therefore, on their suitability for class room presentation. The selection of one particular mode of introduction of this topic depends on the teacher, the class, and the aim of the course being taught. This article is the result of dissatisfaction with all three approaches. The author's teaching experience in this area has been largely drawn from two fields: the educa tion of engineering undergraduates and the late secondary instruction of the dis advantaged. Both these groups experience motiva tional difficulties with the three standard approaches listed above. Approach num ber three, initially the most promising, suffers from the difficulty that condition (6) appears somewhat arbitrary. In searching for a possible new ap proach, the author was guided by the re flection that Napierian logarithms in fact preceded historically the common (base 10) logarithms of Briggs. It is, indeed, a common misconception that Napier discovered natural logarithms. The term Napierian, still used in some table-books as a synonym for natural, no doubt adds to the confusion. It is not the point of this article to describe the history of logarithms. This is already available in the works of Cajori (1913), Boyer (1968, pp. 342-44), Cairns (1928), and others. It is, however, fair to record that the approach described here arose largely from my own reading of these works, although it duplicates none of them. The present approach is numerical to a point, but the computational aspect is used in tandem with the more theoretical view that it motivates. It begins with the Napierian idea that before a table of logarithms can be used, it must first be constructed. The actual construction of a logarithmic table is a problem in numerical analysis. Napier attacked it by constructing a table of powers. To take a suitable base and find its successive powers is much easier than to seek to calculate logarithms to a previously designated base.1