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114 Exponent and Logarithm Problems from the AwesomeMath Summer Program

Titu Andreescu and Sean Elliott
XYZ Press
Publication Date: 
Number of Pages: 
Problem Book
[Reviewed by
Mehdi Hassani
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Exponents and logarithms are very important and useful concepts in mathematics. Many significant mathematical results are explained in terms of logarithms; number theory, for example, is full of surprising facts formulated in terms of logarithms. For this reason, among the many topics covered in mathematical competitions, problems about exponents and logarithms are particularly valuable to work on.

The book under review, as its title indicates, includes some clever problems selected from the “AwesomeMath Summer Program.” The book arranged in ten chapters. Chapters 1–4 contain introductory topics and subjects related to exponents and logarithms, covering the basics, algebraic techniques, equations and systems of equations, and inequalities. Chapters 5 and 6 consider exponents and logarithms in number theory and calculus, and the authors study some problems from both of these viewpoints, comparing the fruitfulness of the techniques. Although the chapters are full of examples, the authors selected 57 introductory problems and 57 advanced problems, mostly from mathematical competitions. These problems are arranged in Chapters 7 and 8, and full solutions are given in Chapters 9 and 10.

There is a curious thing about this book: on the one hand, we find some very tricky problems in it, seemingly hard to attack; on the other hand we realize that everything in the book quite elementary. The solutions for many problems don’t actually need serious preliminary results and facts, hence students at every level from high-school to college will find some interesting and enjoyable problems. Instructors will be able to find some interesting examples to introduce in their teaching of the very classical topic of exponents and logarithms.

Mehdi Hassani is a faculty member at the Department of Mathematics, Zanjan University, Iran. His fields of interest are Elementary, Analytic and Probabilistic Number Theory.