We study the solution sets of several equations of the form  where  and   and  are each one of the basic arithmetic operations; and a and x vary over the set of positive real numbers. For most of the equations studied here, the cardinality of the solution set is determined. As the prerequisites do not go beyond the Intermediate Value Theorem and the Mean Value Theorem, this note could find use as enrichment material in courses on calculus or real analysis.

logarithm, base, identity, equation, solution set, intermediate value theorem, mean value theorem, real-valued function, derivative.

Received: November 6, 2020; Accepted: January 21, 2021; Published: February 22, 2021

How to cite this article: David E. Dobbs, On the real solutions of  and related equations , Far East Journal of Mathematical Education 21(1) (2021), 43-78. DOI: 10.17654/ME021010043

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] S. Banach, Theory of linear operations, Translated from the French by F. Jellett. With comments by A. Pełczyński and Cz. Bessaga, North-Holland Mathematical Library 38 (1987), North-Holland Publishing Co., Amsterdam.
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