## Mathematics, Department of

#### Date of this Version

5-1975

#### Abstract

A "from scratch" proof of the differentiability of *a ^{x}, a* > 0, is avoided by essentially all modern-day authors. A slick and popular way of handling the problem is to define

*a*as

^{x}*e*log

^{x}*a*its differentiability and other properties following from that of the functions

*e*and log

^{x}*x*. Unfortunately, the usual definitions of

*e*and log

^{x}*x*involve relatively sophisticated ideas (e.g., integration or power series). Furthermore, the student, having heard of

*e*, the natural logarithm base, at an early stage of his development, is hardly enlightened when he is told that

*e*is

*e*

^{1}. He would have a much better feeling for the "naturalness" of

*e*if it were defined as that number

*a*for which (

*a*)' =

^{x}*a*.

^{x}The purpose of this note is to provide a direct and relatively simple way of getting at the differentiability of

*a*.

^{x}
## Comments

Published in

The American Mathematical Monthly,Vol. 82, No. 5 (May, 1975), pp. 505-506 Copyright 1975 Mathematical Association of America. Used by permission.