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The Logarithmic Function, and the Numbers e and π
Published online by Cambridge University Press: 03 November 2016
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1. This perennial question has come up, once again, in a Note (No. 1805, February 1945) by the President for the year.
The object of this contribution is to try to state something like a full case (the writer seems to have been working at this for most of his academic life) for the traditional view of the subject, for which the President contends : while at the same time indicating the complementary importance of the other view. (As in all such cases, strong difference of view by competent authorities implies important truth on both sides (which it is necessary to synthesise; and this is, in fact, a fundamental mathematical case of the kind).)
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- Copyright © Mathematical Association 1946
References
page note 132 * See Gazette, XXII, 250, pp. 226-8 (July 1938); and The Number-System of Arithmetic and Algebra (Melbourne University Press), by the writer (in the M.A. Library).
page note 132 † Such a secondary form as logb a = 1/loga b is not elementary, because only one of the two logarithms in question is there admissible.
page note 132 ‡ Most of the workers who use logarithm tables freely have no full knowledge of the difficult theory involved.
page note 132 § The variation of these functions is discussed in Appendix IV of the above-mentioned Number-System (one simple step being inadvertently omitted there).
page note 133 * It is notable that in both cases the variation of the ‘difference’ is expressible in terms of that of the function itself. This is characteristic simplicity of the fundamental.
page note 133 † See § 3, above; also Gazette, XIII, 190, pp. 411-3 (October 1927).
page note 133 ‡ It is easy to reduce the graphical discussion of the gradient-variation to analytical form (in terms of the coordinates of three points V1, V2, V3), and—on the basis of the monotonie variation—to argue that u→0 when x→∞ : it being sufficient to use 1 + x = a n , with n integral and tending to ∞
page note 134 * r(r-1)<2n, for values of r up to a certain value k(<n); when r>k, the inequality for pr is implied by 0<pr< 1 (e.g. if n = 15, k=5).
page note 134 † Upper limit in terms of e itself: another remarkable instance of simplicity of the fundamental.
page note 134 ‡ The particular-simpler-result being remembered, the general result can be ‘recovered’ by using loga x = log x/log a
page note 135 * It is worth noting that “exceptional cases” in mathematics are practically always of positive significance, and are commonly gateways to important developments.
page note 135 † This more general form of these limits has commonly been overlooked (see, e.g. the present writer, Gazette, XIII, 190, p. 414), and the corresponding differentiations deduced from the standard special case.
page note 136 * See Gazette, XXII, 250, pp. 229, 233, for the inter-relation of these two cases, and of the following “decimal” forms-which were taken out by modern calculating machines (used in Munitions work) by Mr. J. A. Macdonald, M.A., a former student of the writer.
page note 136 † (eiπ) for “principal value” : the general “power” form being ab = E(b.L(a)), where L(z) is the “inverse” of E(z), and is equal to log |z| + i Amp (z).