# FW: Re: Math Trivia

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-----Original Message-----
From:	Dukelow, James S Jr
Sent:	Thursday, April 16, 1998 6:17 PM
Subject:	Re: Math Trivia

Well -- actually it isn't so trivial.  Skip Danielson got closest to the reason
we use the "natural" exponential function, e**x, and the "natural" logarithm, ln
x.

If you sit down to calculate the derivative at x of an exponential function,
a**x, for any old a whatsoever, you get:

limit as h-->0 of [ a**(x+h) - a**x ]/h
= lim(h-->0)  a**x * [ a**h - 1 ]/h
= a**x * lim(h-->0) [ a**h - 1 ]/h

The constant  e  is the unique number for which the limit on the right hand end
of the final expression is equal to 1,

WHICH MEANS, with the natural exponential function you get a differentiation and
integration formulas that are not cluttered up with a multiplicative constant
that is not equal to 1.

Similarly, the natural logarithm is the only log function with differentiation
and integration formulas that are not cluttered by an ugly factor ("ugly"
meaning not equal to 1).

Another way of looking at it -- e**x is the only exponential function that
crosses the y-axis with slope = 1 and the natural logarithm is the only log
function crossing the x-axis with slope = 1.

The use of both might be considered a result of the mathematician's natural
laziness, much like when they renormalize the wave equation so that the speed of
light (or whatever) c = 1.

Best regards.

Jim Dukelow
Pacific Northwest National Laboratory
Richland, WA
jim.dukelow@pnl.gov
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