A Radical Approach to the Logarithm

By David Bressoud @dbressoud

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My experience has been that most students never understand logarithms. Inevitably they struggle to remember that log(a*b) = log(a) + log(b) and log(a^n) = n log(a). The standard approach is to introduce the logarithmic function as the inverse of the exponential function. That gives the instructor confidence that the rule for the logarithm of a product is fully justified, even if that explanation does not always stick with students. For those insisting on full “rigor,” the logarithm is defined in terms of the definite integral of the reciprocal of x. That produces an even less memorable explanation of how the logarithm turns products into sums.

For me, this raises the question: Why not begin with an exploration of what it would mean to build a function that turns products into sums? After all, this is where Napier began a century and half before Euler would establish the exponential as an actual function well-defined at all real values. I am not going to be crazy enough to try to explain Napier’s original creation. The interested reader can find it laid out in all of its gory detail in Chapter 3 of Julian Havil’s magnificent book John Napier: Life, Logarithms, and Legacy. But I would like to start with a naïve search for a function that turns products into sums. I then will draw inspiration from Leonard Euler’s Introduction to Analysis of the Infinite to create the natural logarithm.

Let us quite arbitrarily decide to designate this function by the letter L, L(a*b) = L(a) + L(b). Our function L must map the multiplicative identity to the additive identity: L(1) = 0. Our function will be uniquely determined by the choice of two constants I call q and h for which multiplication by q gets mapped to addition by h. That is to say, if L(x) = y, then

L(xq) = y + h = L(x) +L(q).

Napier referred to these functions as logarithms. He never explained how he came up with this name. It is clearly a combination of the Greek words logos and arithmos. Arithmos is not particularly problematic. It is the root of our word arithmetic and can refer to calculation. Logos is more problematic. The common etymological explanation is that it refers to ratio, which appears to be the sense in which it was used Apollonius (Heath, 1981b, p. 175). These were calculations involving ratios. But logistiki can also refer to calculation in the sense of practical mathematical calculations, from which it became the root for logistics. Arithmetiki in contrast refers to the theory of numbers. Heath explains that this is how Plato thought of these terms (Heath, 1981a, pp. 13–14). Napier’s choice of logarithm might well have been a means of explaining his creation as encapsulating the theory of numbers applied to calculations. That seems much more satisfying than “ratio-numbers.”

How to choose q and h? We begin by considering the average rate of change of the logarithm over the interval [x,xq],

(L(x) – L(xq)/(x – xq) = –L(q)/(x – xq) = (1/x)*h/(q–1).

This is interesting. The average rate of change is always equal to the reciprocal of the value at one end of the interval multiplied by a constant that depends on q and h. Let us call the reciprocal of that constant k = k(q,h),

k = (q–1)/h.

It would be very nice if the derivative of L(x) were just 1/x. That is to say, we would like the limit as q approaches 1 and h approaches 0 of k(q,h) to be 1, but we have a problem. Setting q = 1 and h = 0 yields k = 0/0. Here is where we turn to a trick introduced by Euler.

We solve for q in terms of k and h,

q = 1+kh.

Rather than setting q = 1, we let u be infinitely small and consider

q^u = (1+kh)^u.

Since u is infinitely small, we can replace the function of h on the right-hand side by its linear approximation at h = 0,

q^u = 1+ukh.

Now pick an arbitrary positive value z and define j to be the infinitely large number with the property that ju = z.

On the right-hand side, we replace u by z/j

Remember that k is the reciprocal of the constant that multiplies 1/x in the derivative of L(x), and we want k = 1. At this point we can simply set k equal to 1 and then recognize that for infinitely large values of j, (j–1) (j–2)…(j–n) /j^n = 1.

Setting z = 1 gives us the relationship between q and h when the derivative of L(x) equals 1/x. We call this particular logarithm the natural logarithm.

Since L(qz) = zh, it is useful to find the value of q when h = 1. We denote this specific value of q by the letter e.

We have established that when L maps products to sums and its derivative is simply 1/x, then

L(e^z) = z.

In other words, the natural logarithm L is the inverse function of the exponential that maps z to e^z. We wound up where most treatments of the natural logarithm begin, but we started with the assumption that this function would take products to sums, and along the way we found the exact value of its base.

References

Euler, L. (2008). Introduction to Analysis of the Infinite, vol I. New York, NY: Springer Verlag.

Havil, J. (2014) John Napier: Life, Logarithms, and Legacy. Princeton, NJ: Princeton University Press.

Heath, T. (1981a). A History of Greek Mathematics, vol. I. New York, NY: Dover Publications.

Heath, T. (1981b). A History of Greek Mathematics, vol. II. New York, NY: Dover Publications.

David Bressoud is DeWitt Wallace Professor Emeritus at Macalester College and former Director of the Conference Board of the Mathematical Sciences. davidbressoud.org

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