Logarithms: The Early History of a Familiar Function - Logarithms: A 'Great Tale' for Use in the Classroom

The logarithmic relation, captured in modern symbolic notation as \[ \log(a\cdot b) = \log(a) + \log(b),\] is useful primarily because of its power to reduce multiplication and division to the less involved operations of addition and subtraction. When this relation hit the scene in the early seventeenth century, its impact was substantial and immediate. Modern historians of mathematics, John Fauvel and Jan van Maanen (2000), illustrate this vividly:

When the English mathematician Henry Briggs learned in 1616 of the invention of logarithms by John Napier, he determined to travel the four hundred miles north to Edinburgh to meet the discoverer and talk to him in person. (p. xi)

Indeed, Fauvel and van Maanen assert that “the meeting of Briggs and Napier is one of the great tales in the history of mathematics” (p. xi). Unfortunately, it seems that many teachers (and their students) are not aware of this particular “great tale” – or, at most, superficially associate the names Briggs and Napier with the invention of the logarithm. Typically, these groups know little about the original conceptions of the logarithmic relation.

An anecdote concerning a conversation between Fauvel (1995) and a colleague reinforces this. Fauvel recounted that when he inquired of his colleague how to teach logarithms, the colleague responded, “Whatever for? Surely no one needs to learn about those any more, now that we have calculators and computers” (p. 39). Many teachers, when approached about the possibility of teaching logarithms using a historical context, may express the same opinion. Fauvel's counter-argument to his colleague was that “logarithms are a good and accessible example of something fundamentally changing its conceptual role within mathematics” (p. 45). Indeed, examining the historical development of the logarithm with students by exploring arithmetic and geometric progressions allows students “a more deeply rooted understanding of what is going on” (p. 42).

The modern concept of the logarithm typically appears late in a second algebra or precalculus course (grades 10 or 11 in the US), situated after a study of polynomial and rational functions, but before sequences and series and conic sections. This organization often leaves students with an impression of disconnectedness between mathematical topics. Complicating matters further, the logarithm is often presented only briefly in such an algebra or precalculus course, in order to lead to a broader study of logarithmic functions to match students' previous study of other functions (e.g., linear, polynomial, rational). Lastly, in order to focus on the study of logarithmic functions, instruction and curricular organization dictate that this function exist as the inverse of the exponential function. This, in particular, contrasts starkly with the historical circumstances: in fact, the trajectory of modern mathematical pedagogy does not imitate history, as exponentials arrived on the mathematical scene well after the introduction of the logarithm!

Victor Katz (1995; 1997) provided a succinct argument for examining the development of logarithms from a historical perspective. He observed that Napier developed logarithms “for use in the extensive plane and spherical trigonometrical calculations necessary for astronomy” (Katz, 1995, p. 49). Although the motivation for developing logarithms is significant, Katz noted that, in general, students today often know very little about astronomy and about the magnitude of both the numbers and the calculations involving such numbers that were necessary to advance the science of astronomy. Astronomical advances have remained critical throughout civilization, however, and Katz (1997) indicated that, “it is well for us to introduce it [astronomy] whenever possible” (p. 63).

Although there is a strong temptation simply to present the definition and several properties of the logarithm and exercises to practice each, we propose that incorporating original and parallel insights of the logarithm can enrich instruction and learning of the topic, both for this concept and more broadly for a student's understanding of mathematics and its relations and development.