# [seqfan] Re: 2(x - 1)^x = x^x

Neil Sloane njasloane at gmail.com
Fri Mar 2 14:30:40 CET 2018

```> shall we also have the decimal expansion of the number discussed here as a
sequence?

Of course!

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Email: njasloane at gmail.com

On Fri, Mar 2, 2018 at 7:05 AM, Hugo Pfoertner <yae9911 at gmail.com> wrote:

> Given that we get hits like  "Decimal expansion of log_10(19)"
> https://oeis.org/A155062 by a search
> https://oeis.org/search?q=7%2C2%2C2%2C8%2C9&sort=&language=&go=Search
> shall we also have the decimal expansion of the number discussed here as a
> sequence?
>
> On Thu, Mar 1, 2018 at 3:51 PM, <rgwv at rgwv.com> wrote:
>
> > And that is what I got yesterday. RGWv
> >
> > -----Original Message-----
> > From: SeqFan <seqfan-bounces at list.seqfan.eu> On Behalf Of neil greubel
> > Sent: Wednesday, 28 February, 2018 11:53 PM
> > To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> > Subject: [seqfan] Re: 2(x - 1)^x = x^x
> >
> > The exact solution to x^x = 2*(1-x)^x is x = 1 / (1 + W(ln2/2)/ln2) ,
> > where W(x) is the Lambert W-function.
> >
> > Proof: By taking the ln of both sides of the equation x lnx = ln2 + x
> > - (ln2/x) = ln((1-x)/x)
> > exp(- ln2/x) = (1-x)/x
> > (1/2) = (1/x - 1) * exp( ln2 / x) * (1/2)
> > (ln2 / 2) = ln2 * (1/x - 1) * exp(ln2 / x - ln2) =>
> > ln2 *(1/x - 1) = W(ln2 / 2)
> > 1/x = 1 + W(ln2 / 2) / ln2
> > and finally
> > x = ln2 / (ln2 + W(ln2/2)) = 0.72289208118314779553.....
> >
> > On Wed, Feb 28, 2018 at 11:17 PM, <rgwv at rgwv.com> wrote:
> >
> > > 0.722892081183147795532365351219771674355877736768641720125436
> > > 5151991334348752469431309465384425833668792370029595892746086...
> > >
> > > -----Original Message-----
> > > From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of
> > > WALTER KEHOWSKI
> > > Sent: Wednesday, 28 February, 2018 6:18 PM
> > > To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>; юрий
> > > герасимов <2stepan at rambler.ru>
> > > Subject: [seqfan] Re: 2(x - 1)^x = x^x
> > >
> > > The equation 2(1-x)^x = x^x has the zero x=0.7228920812. HTH.
> > >
> > > > On February 26, 2018 at 2:32 AM юрий герасимов <2stepan at rambler.ru>
> > > wrote:
> > > >
> > > >
> > > > Dear SeqFans.
> > > > Help find the first members of this sequence:
> > > > Decimal expansion of the real positive solution to 2(x - 1)^x = x^x.
> > > > Thanks. JSG.
> > > >
> > > > --
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> > >
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> > >
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> > >
> >
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>
```