Abstract
Exponential and logarithmic functions (and in general all transcendental functions) can be analyzed by developing inequalities that compare them with polynomial and rational functions. This method lies in the heart of calculus as advocated by Euler, Newton, Leibniz, the Bernoulli brothers, Taylor, and others. The most prominent applications of these inequalities are the existence and convexity properties of the exponential and logarithmic functions. We present here the two principal approaches, Newton’s and Euler’s, with full details. We use the method of means (Section 3.2) to derive the power series expansion of the natural exponential function without calculus. An optional section derives explicit formulas for all power sums (introduced in Section 3.2) in terms of the Bernoulli numbers. This chapter is concluded by presenting sharp estimates on the sum of reciprocals of the first n natural numbers, and a large variety of sophisticated but lesser-known limits involving natural exponents and logarithms.
“A Scottish baron has started up, his nameI cannot remember, 1 but he has put forthsome wonderful mode by which all necessityof multiplications and divisions are commutedto mere additions and subtractions.” Johannes Kepler, from a letter to Wilhelm Schickard, 2 upon having seen a copy of Napier’s Mirifici Logarithmorum Canonis Descriptio
(Description of the Admirable Cannon of Logarithms).
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Notes
- 1.
John Napier of Merchiston (1550–1617).
- 2.
In 1617, the year of Napier’s death.
- 3.
For two different proofs, see the author’s Glimpses of Algebra and Geometry, 2nd ed. Springer, New York, 2002.
- 4.
This example needs a computer algebra system.
- 5.
This is our s 10(1001).
- 6.
See also History in Section 10.5.
- 7.
This was also a problem (including negative integers) in the William Lowell Putnam Exam, 1960.
- 8.
This is due to Virgil Nicula.
- 9.
We will treat this is more detail in Section 10.5.
- 10.
For two sequences \((a_n)_{n\in \mathbb {N}}\) and \((b_n)_{n\in \mathbb {N}}\) with non-zero terms, we write a n ∼ b n as n →∞ if limn→∞ a n∕b n = 1.
- 11.
Note the extended range of p as a special case of Example 3.4.1, and also the moved up value of n to n + 1.
- 12.
See also Kubelka, R.P., Means to an end, Math. Mag. 74 (2001) 141–142, and Conway Xu, A GM-AM ratio, Math. Mag. 83 (2010) 49–50.
- 13.
This is due to the Roumanian mathematician Traian Lalescu (1882–1929).
- 14.
This is due to the Roumanian mathematician D.M. Bǎtineţu-Giurgiu (1936-).
- 15.
This generalization and the next example are due to Virgil Nicula.
- 16.
A special case was a problem in the American Mathematics Competition, 2015.
- 17.
This was Problem 5 in Round 1 and Year 32 of the USA Mathematical Talent Search; Academic Year 2020/2021.
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Toth, G. (2021). Exponential and Logarithmic Functions. In: Elements of Mathematics. Undergraduate Texts in Mathematics(). Springer, Cham. https://doi.org/10.1007/978-3-030-75051-0_10
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DOI: https://doi.org/10.1007/978-3-030-75051-0_10
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