Solving an Exponential Equation
Date: 04/02/2001 at 10:52:24 From: Joe Moore Subject: Solution to an exponential equation Solve for x: x^3 = 2^x I have created a computer program to solve this equation using the brute force method. x is close to 1.373468. I have tried logarithms, factoring, derivatives, and limits, but I can't isolate x. Thanks for any help.
Date: 04/02/2001 at 16:11:01 From: Doctor Rob Subject: Re: Solution to an exponential equation Thanks for writing to Ask Dr. Math, Joe. You are right; you can't isolate x. The best you can do is what you have already done: solve numerically. The answer, to more decimal places, is x = 1.373467119696165166721660004192166... There is an unfamiliar function called Lambert's W-function that allows you to isolate x. W(y) is defined to be the function such that W(y)*e^W(y) = y You can verify that x = e^(-W[-ln(2)/3]) satisfies your equation. Of course that is not helpful, because there isn't a good way to compute the values of the W-function aside from the numerical ones. This function is available on the Maple(TM) computer algebra system. For the W-function, see Frequently Asked Questions in Mathematics, from the Sci.Math FAQ team: Name for f(x)^f(x) = x http://db.uwaterloo.ca/~alopez-o/math-faq/node42.html - Doctor Rob, The Math Forum http://mathforum.org/dr.math/
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