It may come as a surprise to many that often times mathematical concepts don't end up like they started! For those of you who think mathematics is timeless, fixed, and full of unchanging truths, such a proposition may seem unbelievable. But there are many instances in the history of mathematics of the development of a mathematical concept way beyond the purposes and potentialities that its original inventors intended. An example that will be familiar to you all is the logarithm.

What is a logarithm? Ask a modern mathematician nowadays and you will get a very different answer from the one you might have got from a mathematician several centuries ago. Indeed, even the very first mathematicians who worked with the logarithmic relation would have given an explanation that would seem quite foreign to a modern mathematician. So how did the logarithmic relation come about, and how is it that the concept underwent so much change? We will address these questions by looking at the emergence of this concept, and examining some of the issues surrounding its origins.

In fact, the question of the origins of the logarithmic relation does not have a simple answer. At least two scholars, the Scottish baron John Napier (1550-1617) and Swiss craftsman Joost Bürgi (1552-1632), produced independently systems that embodied the logarithmic relation and, within years of one another, produced tables for its use. This parallel insight is fascinating and rich in historical detail, and it reveals some methodological challenges for historians of mathematics. In light of all this, we will examine the ideas of these two scholars, as well as explore how historians have portrayed this intricate situation and the questions it raises about mathematics.

In large part, we intend to re-introduce teachers to a concept that is often taught without any reference to its original appearance on the mathematical scene. We hope that a close examination of Napier's and Bürgi's conceptions will enable teachers to consider alternative placement for introducing the idea of logarithms – as part of or after a unit on sequences. Furthermore, we provide in what follows mathematical and historical content, as well as student exercises, to promote the teaching of the logarithmic relation from its historical roots, which are firmly situated in simultaneous consideration of arithmetic and geometric sequences.