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*To*: "'radsafe@romulus.ehs.uiuc.edu'" <radsafe@romulus.ehs.uiuc.edu>*Subject*: FW: Re: Math Trivia*From*: "Dukelow, James S Jr" <jim.dukelow@pnl.gov>*Date*: Thu, 16 Apr 1998 18:20:42 -0700

-----Original Message----- From: Dukelow, James S Jr Sent: Thursday, April 16, 1998 6:17 PM To: 'radsafe@romulus.ehs.uiuc.edu' 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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