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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   

- Doctor Rob, The Math Forum   
Associated Topics:
High School Functions

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