Glog

In: Science

Submitted By richtalk
Words 3007
Pages 13
A Generalized Logarithm for
Exponential-Linear Equations
Dan Kalman

Dan Kalman (kalman@email.cas.american.edu) joined the mathematics faculty at American University in 1993, following an eight year stint in the aerospace industry and earlier teaching positions in Wisconsin and South Dakota.
He has won three MAA writing awards, is an Associate
Editor of Mathematics Magazine, and served a term as
Associate Executive Director of the MAA. His interests include matrix algebra, curriculum development, and interactive computer environments for exploring mathematics, especially using Mathwright software.

How do you solve the equation
1.6x = 5054.4 − 122.35x?

(1)

We will refer to equations of this type, with an exponential expression on one side and a linear one on the other, as exponential-linear equations. Numerical approaches such as Newton’s method or bisection quickly lead to accurate approximate solutions of exponential-linear equations. But in terms of the elementary functions of calculus and college algebra, there is no analytic solution.
One approach to remedying this situation is to introduce a special function designed to solve exponential-linear equations. Quadratic equations, by way of analogy, are

solvable in terms of the special function x, which in turn is simply the inverse of a very special and simple quadratic function. Similarly, exponential equations are solvable in terms of the natural logarithm log, and that too is the inverse of a very special function. So it is reasonable to ask whether there is a special function in terms of which exponential-linear equations might be solved. Furthermore, an obvious strategy for finding such a function is to invert some simple function connected with exponentiallinear equations.
This line of thinking proves to be immediately successful, and leads to a function I call glog…...

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