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, 7:28 pm

Newsgroups: sci.math

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

Date: Wed, 11 Apr 2007 02:28:59 +0300

Local: Tues, Apr 10 2007 7:28 pm

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

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"David W. Cantrell" <DWCantr...@sigmaxi.net> wrote in message
news:20070410190915.721$e0@newsreader.com...

> Gerry Myerson <ge...@maths.mq.edi.ai.i2u4email> wrote:
> > In article <1176239376.759713@athprx04>,
> > "Ioannis" <morph...@olympus.mons> 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)

> > 2^x is smaller than x^{12} at x = -1,
> > bigger at x = 0,
> > smaller at x = 2,
> > and bigger at x = 84 ( 2^{84} = (2^7)^{12} = 128^{12} > 84^{12} ),
> > so three real solutions.

> Yes indeed. The two roots given by Ioannis are obtained using the principal
> branch of the Lambert W function. The third root is given by

> x = -12*W_{-1}(-log(2)/12)/log(2)

> where W_{-1} denotes the -1 branch.

Thanks to both for spotting the omission. I did an evalf(") on the exact
solutions Maple provided to scan for real solutions, but I missed that one,
which seems to be close to x_3 ~= 74.66932553

Moral of the story: When an equation is solvable exactly by W, *always* check
the -1 branch as well :-)

> David

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

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