>>> But that's beside the point. By defining log x as the inverse >>> function of 10^x or e^x you are just postponing the problem. How >>> do you define 10^x? Don't tell me that's defined in terms of >>> logarithms :-)
I doubt that when Napier invented logarithms he did not know the meaning of a^x for a positive and x real. He might not have used the epsilon-delta notation, but he certainly had the idea. Tabulation and interpolation were the methods used BC (before calculus).
I recall having read that logarithms to base 10 were first calculated by computing 10^x for x of the form m/2^n and inverting. Also, Napier first computed logarithms to the base .9999999 by multiplication.
>>> After you clean up your definition of 10^x, stand back and >>> see if that is any more appealing than the definition of ln x >>> as the antiderivative of 1/x.
>>I've always found it humorous that calculus texts start getting sweaty with >>rigor on the derivatives of the exponential functions b^x (b > 0) and their >>companions, the logarithms. Do these students even know, with rigor, what >>sqrt(x) is? Usually not, but they've been merrily differentiating it for >>weeks.
That students do not know the meaning of powers and roots is abysmal. But with the current emphasis on calculation, not on understanding, this is not that surprising. Mathematical concepts need to be taught early, NOT after computation. The students who learn how to carry out calculations of derivatives and antiderivatives are not in any better position to understand what they mean than they were initially, and I believe they are in a worse shape.
>I agree with your assessment of the situation. A good part of the >fine points of calculus is lost on the majority of students.
Of course they are lost. If the concepts of limit and derivative are "done" in one lecture, with one homework assignment, and never appear on final examinations, what do you expect? The impression is given that it is not of much importance. Rather, it is the drill in calculation which should be considered of little importance, from elementary school on. It is useless to know how to add if one does not know what addition means. They do not know even what the properties of the integers are when they take calculus.
>Back to the original point of this thread: I don't want to give the >incorrect impression that I am rigidly set against defining ln x as >the inverse function of e^x.
This is how it was done from around 1600 until almost the present.
It wouldn't hold it against a calculus >book that does. I will be even happier if both approaches -- i.e. >defining e^x in terms of ln x and visa versa -- were presented, possibly >one version in the main text and the other version as an exercise.
One problem with starting with the logarithm as an integral is the problem of handling numbers less than 1. On the other hand, showing that (1+x/n)^n is increasing in n for x > 0 by using the binomial theorem and termwise comparison, and proceeding from there by using properties of limits and infinite series, is, I believe, quite within the range of anyone who has had a modicum of high school algebra and can think. There is no point of delaying it until later. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 firstname.lastname@example.org Phone: (765)494-6054 FAX: (765)494-0558