EXPONENTIALS AND LOGARITHMS BY CONTINUOUS EXTENSION

The exponential and logarithm functions are extraordinarily important in one-variable calculus. However, they are hard to teach with any attempt at rigor without completely confusing students. Conventional approaches are either very roundabout, or skip the hard (and interesting) bits altogether. This paper attempts to bridge the gap. In the process, it discusses a fundamental principle underlying the use of graphing calculators (or computers) —“connecting the dots”, or continuous extension from a dense subset. The challenge (as well as the beauty) of the whole subject is that the exponential and logarithm functions1 play a number of roles in calculus: (R1) bx is the extension of the formula bp/q = q√bp to real x. (R2) ex is its own derivative; e is “some crazy number ” which makes this happen. (R3) bx is proportional to its own derivative; the proportionality factor is lnb. (R4) ex and lnx are inverses, and bx = e(lnb)x. (R5) lnx = ∫ x1 t−1dt. (R6) ex is the solution to the differential equation y ′ = y, y(0) = 1. (R7) ex = limn→∞(1+ x/n)n