roots of x^12 = 2^x - sci.math

Message from discussion roots of x^12 = 2^x

Ioannis

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More options Apr 10 2007, 5:09 pm

Newsgroups: sci.math

From: "Ioannis" <morph...@olympus.mons>

Date: Wed, 11 Apr 2007 00:09:35 +0300

Local: Tues, Apr 10 2007 5:09 pm

Subject: Re: roots of x^12 = 2^x

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"chapkovski" <chapkov...@gmail.com> wrote in message

news:1176236640.212574.296720@a30g2000cwd.googlegroups.com...

> how many roots does this equation have?

Two real ones, approximately at x_0 ~= -.9467803304 and at x_1 ~= 1.063346831,
given by Lambert's W function as:

x = -12*W((+/-) log(2)/12)/log(2)

If you are looking for complex roots, there are more, given by more
complicated exressions in terms of the same function. Briefly, the equation
can be solved using Lambert's W function as follows:

x^12 = 2^x =>
x = (+/-) 2^(x/12) =>
(+/-) x*2^(-x/12) = 1 =>
(+/-) x*exp(-log(2)/12*x) = 1 =>
(+/-) -log(2)/12*x*exp(-log(2)/12*x) = -log(2)/12 =>
x = -12*W((+/-) log(2)/12)/log(2)

To learn more about Lambert's W function, Google it.

> Thanks in advance for explaining

--
I.N. Galidakis --- http://ioannis.virtualcomposer2000.com/

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