9
Exponential and logarithmic functions
9.1 Exponentials and logarithms At least a modicum of basic
information about exponentials is possessed by everyone who looks at
Chapter 9 of a calculus book. Everyone knows that 3" = 3.3.3.3 and
334 _ N/T Such formulas as
as` c ab = (/)5,
in which it is supposed that a > 0, are familiar. Remark 20 which
appears among the problems at the end of this section provides basic
theory of a= for the case in which a > 0 and the exponent x is a rational
number, that is, x = P/Q, where P and Q are integers for which Q ,E 0.
There is a reason why this theory is often neglected. It is too difficult
for elementary algebra books because it requires use of the intermediate
value theorem and hence requires knowledge of completeness of the real-
number system. It is too simple for calculus books because it is mostly
elementary algebra. The mathematical theory of exponents, like the
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