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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
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])

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/
```
Associated Topics:
High School Functions

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