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

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

Zdislav V. Kovarik

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More options Apr 11 2007, 12:24 pm

Newsgroups: sci.math

From: "Zdislav V. Kovarik" <kova...@mcmaster.ca>

Date: Wed, 11 Apr 2007 12:24:57 -0400

Local: Wed, Apr 11 2007 12:24 pm

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

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On Wed, 11 Apr 2007, Ioannis wrote:
> "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)

Another real root, between 74 and 75, got missed:

-12*lambertw(-1,-1/12*log(2))/log(2)

Cheers, ZVK(Slavek)
[Nothing more added by me]

> 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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