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Re: Definition of the exponential and natural logarithm functions [was....doozi]
Posted:
Oct 17, 2001 4:05 PM


On Mon, 15 Oct Lou Talman wrote in part:
:There are three principal methods for giving rigorous definitions of the :exponential/logarithm pair of functions: : : 1. Definite integral : 2. Series : 3. Extension by continuity.
Listowner Flashman replies:
: A fourth approach to the definition of the natural exponential function : can be based on the differential equation y'=y with y(0)=1  motivated by : population growth models, amd made credible using a mixture of : differential equation heuristics and properties of solutions to DE's that : are accessible to first year calculus students.
The operative word here is *rigorous*. There certainly is a rigorous theory of differential equations, based on existence and uniqueness theorems. It would be somewhat misguided to introduce freshmen to these things.
An axiomatic approach, in contrast, based on, say, a functional equation and continuity, obviates the need for developing machinery, and encapsulates the fundamental facts based from which the properties of the function are derived.
However, if rigor is not an important consideration (and of course there are definitions of rigor that are very forgiving indeed) then I would say the differential equation approach is at least on a par with the other three.
It is almost impossible to predict which of these approaches will be most attractive to a given set of students!
P.S. How about the definition exp(x) = lim( {1 + x/n}^n, n > infinity}?
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