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roots of x^12 = 2^x

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chapkovski

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Apr 10, 2007, 4:24:00 PM4/10/07
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how many roots does this equation have?

Thanks in advance for explaining

smn

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Apr 10, 2007, 4:56:45 PM4/10/07
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On Apr 10, 1:24 pm, "chapkovski" <chapkov...@gmail.com> wrote:
> how many roots does this equation have?
>
> Thanks in advance for explaining

For x<= 0 look at the function y=2^x-x^12 which is strictly
increasing from -00 (atx= -00) to 1 at x=0 .Therefore
there is one root here.

for x>0 try to solve x^12/2^x=1 .take the natural logarithm(ln) to
reach the equivalent equation (lnx)/x=(ln2)/2 .The graph of y=(lnx)/x
is easily sketched from calculus from which you can read off the
number of roots.smn

Ioannis

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Apr 10, 2007, 5:09:35 PM4/10/07
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"chapkovski" <chapk...@gmail.com> wrote in message
news:1176236640.2...@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/

Gerry Myerson

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Apr 10, 2007, 6:48:27 PM4/10/07
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In article <1176239376.759713@athprx04>,
"Ioannis" <morp...@olympus.mons> wrote:

> "chapkovski" <chapk...@gmail.com> wrote in message
> news:1176236640.2...@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.

--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)

David W. Cantrell

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Apr 10, 2007, 7:09:12 PM4/10/07
to

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.

David

Ioannis

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Apr 10, 2007, 7:28:59 PM4/10/07
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"David W. Cantrell" <DWCan...@sigmaxi.net> wrote in message
news:20070410190915.721$e...@newsreader.com...

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

Phil Carmody

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Apr 11, 2007, 9:26:51 AM4/11/07
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"Ioannis" <morp...@olympus.mons> writes:
> "David W. Cantrell" <DWCan...@sigmaxi.net> wrote in message
> news:20070410190915.721$e...@newsreader.com...
> >
> > Gerry Myerson <ge...@maths.mq.edi.ai.i2u4email> wrote:
> > > In article <1176239376.759713@athprx04>,
> > > "Ioannis" <morp...@olympus.mons> wrote:
> > >
> > > > "chapkovski" <chapk...@gmail.com> wrote in message
> > > > news:1176236640.2...@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)

> 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 :-)

Not at all. The moral of the story is to not wave great hefty tools
like Lambert W functions around when techniques that the average
10 year old should know provide more insight into the solution.

I am aghast that you seemed to think that x^12 would dominate 2^x
as x increased.

Phil
--
"Home taping is killing big business profits. We left this side blank
so you can help." -- Dead Kennedys, written upon the B-side of tapes of
/In God We Trust, Inc./.

Ioannis

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Apr 11, 2007, 11:07:52 AM4/11/07
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"Phil Carmody" <thefatphi...@yahoo.co.uk> wrote in message
news:871wirc...@nonospaz.fatphil.org...
[snip]

> > Moral of the story: When an equation is solvable exactly by W, *always*
check
> > the -1 branch as well :-)
>
> Not at all. The moral of the story is to not wave great hefty tools
> like Lambert W functions around when techniques that the average
> 10 year old should know provide more insight into the solution.

Nah. I was just too bored to perform a full analysis of the equation.

> I am aghast that you seemed to think that x^12 would dominate 2^x
> as x increased.

It wasn't my fault. Honest :-) That was Maple V release 4's fault. To make
sure that I got the right behavior at +oo, I tried a quick:

> plot(x^12-2^x,x=-infinity..infinity);

Unfortunately, version V of Maple misbehaves on this graph, producing slightly
different variants each time. The time I checked, it produced a spike at +oo
for x->oo (erroneous), so I didn't check further. By breaking the domain of
the graph into smaller regions one can get the correct behavior, but I
stupidly relied on just that one glance I got from the -infinity..infinity
range, which was incorrect.

Shows the dangers of relying too much on CAS for analyses. Maybe we should ban
W altogether...

> Phil
> --

Zdislav V. Kovarik

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Apr 11, 2007, 12:20:25 PM4/11/07
to

On Tue, 10 Apr 2007, chapkovski wrote:

> how many roots does this equation have?
>
> Thanks in advance for explaining

Three real roots: look at the (real-variable) equivalent equation

x^2 - 2^(x/6) = 0,

use repeatedly Rolle's Theorem to show that it cannot have more than
three roots, and use Intermediate Value Theorem to locate one root at a
time:

between (-1) and 0,
between 1 and 2,
and a large positive one because 2^(x/6) eventually outgrows x^2.
(The location is: between 74 and 75).

Non-real roots: there are infinitely many of those because Lambert's
W-function has infinitely many branches, and on top of that, the twelfth
root function has 12 branches.

My computer equation solver found ten more non-real roots, in the strip
-1 < Re(x) < 1. But that is no promise that other roots will stay in that
strip - actually there is a root close to 121.55+111.31i.

Read about Lambert's W-function. The smaller of the positive roots was
found to be (log means the natural logarithm)

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

I may have answered more than you required.

Cheers, ZVK(Slavek).

Zdislav V. Kovarik

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Apr 11, 2007, 12:24:57 PM4/11/07
to

On Wed, 11 Apr 2007, Ioannis wrote:

> "chapkovski" <chapk...@gmail.com> wrote in message
> news:1176236640.2...@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]

Ioannis

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Apr 11, 2007, 6:07:42 PM4/11/07
to
"Ioannis" <morp...@olympus.mons> wrote in message
news:1176304073.615854@athnrd02...

>
> "Phil Carmody" <thefatphi...@yahoo.co.uk> wrote in message
> news:871wirc...@nonospaz.fatphil.org...
> [snip]
[snip]

> > I am aghast that you seemed to think that x^12 would dominate 2^x
> > as x increased.
>
> It wasn't my fault. Honest :-) That was Maple V release 4's fault. To make
> sure that I got the right behavior at +oo, I tried a quick:
>
> > plot(x^12-2^x,x=-infinity..infinity);
>
> Unfortunately, version V of Maple misbehaves on this graph, producing
slightly
> different variants each time. The time I checked, it produced a spike at +oo
> for x->oo (erroneous), so I didn't check further. By breaking the domain of
> the graph into smaller regions one can get the correct behavior, but I
> stupidly relied on just that one glance I got from the -infinity..infinity
> range, which was incorrect.

And indeed, to further my defense for being a careless putz (8*(, here are the
two graphs:

>plot(x^12-2^x,x=-infinity..infinity);

Maple V release 4:
http://misc.virtualcomposer2000.com/graph1.gif

Maple 9:
http://misc.virtualcomposer2000.com/graph2.gif

The Maple V graph is clearly wrong. It's completely missing the third root.
That's the one I saw first and didn't even think twice about exp(x) > x^n,
which is an immediate givaway.

The Maple 9 graph also seems to be wrong, although "less wrong" than the Maple
V graph. I didn't check this one when I answered the question, but to me it
implies that the function reaches +infinity between the second and third root,
and that the third root is very large in magnitude, almost close to +infinity.

It would be nice to see the corresponding graph Maple 10 produces.

David W. Cantrell

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Apr 11, 2007, 8:58:43 PM4/11/07
to
"Ioannis" <morp...@olympus.mons> wrote:
> "Ioannis" <morp...@olympus.mons> wrote in message
> news:1176304073.615854@athnrd02...
> > "Phil Carmody" <thefatphi...@yahoo.co.uk> wrote in message
> > news:871wirc...@nonospaz.fatphil.org...
> [snip]
> > > I am aghast that you seemed to think that x^12 would dominate 2^x
> > > as x increased.
> >
> > It wasn't my fault. Honest :-) That was Maple V release 4's fault. To
> > make sure that I got the right behavior at +oo, I tried a quick:
> >
> > > plot(x^12-2^x,x=-infinity..infinity);

Very interesting. Mathematica, for example, balks at any request to graph
from -Infinity to Infinity.

> > Unfortunately, version V of Maple misbehaves on this graph, producing
> > slightly different variants each time. The time I checked, it produced
> > a spike at +oo for x->oo (erroneous), so I didn't check further. By
> > breaking the domain of the graph into smaller regions one can get the
> > correct behavior, but I stupidly relied on just that one glance I got
> > from the -infinity..infinity range, which was incorrect.
>
> And indeed, to further my defense for being a careless putz (8*(, here
> are the two graphs:
>
> >plot(x^12-2^x,x=-infinity..infinity);
>
> Maple V release 4:
> http://misc.virtualcomposer2000.com/graph1.gif
>
> Maple 9:
> http://misc.virtualcomposer2000.com/graph2.gif
>
> The Maple V graph is clearly wrong. It's completely missing the third
> root. That's the one I saw first and didn't even think twice about
> exp(x) > x^n, which is an immediate givaway.

This probably deserves a new thread, so I've changed the title
to "Comprehensive Graphing".

I've used such graphs for many years (although I did not produce such a
graph for x^12 - 2^x until after you had posted the graphs produced by
Maple). I have called them "comprehensive graphs". Is there an "accepted"
name for them?

For comparison with your Maple 9 graph, here's a comprehensive graph which
I produced with Mathematica:

<http://img264.imageshack.us/img264/4936/comprehensivegraphmq6.gif>

The code I used was

f[x_] := x/(1 - Abs[x]); fInv[x_] := x/(1 + Abs[x]); g[x_] := x^12 - 2^x;
Plot[fInv[g[f[x]]], {x, -1, 1}, AspectRatio -> Automatic, Ticks -> False]

Above, you can see the scaling function f which I used. It's my "default"
for comprehensive graphing, but of course many other such functions could
be used (e.g. Tanh for f, ArcTanh for fInv). What scaling function does
Maple use by default for such graphs? Can the user specify alternative
scaling functions?

> The Maple 9 graph also seems to be wrong, although "less wrong" than the
> Maple V graph. I didn't check this one when I answered the question, but
> to me it implies that the function reaches +infinity between the second
> and third root,

It doesn't imply that to me. Rather, it just shows that the function gets
very large between the second and third roots.

> and that the third root is very large in magnitude,
> almost close to +infinity.

Well, that's presumably because it is! In my comprehensive graph, for
example, the distance between 0 and +oo is 1, while the distance between
that third root and +oo is merely 0.013...

David

Ioannis

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Apr 11, 2007, 9:52:15 PM4/11/07
to
"David W. Cantrell" <DWCan...@sigmaxi.net> wrote in message
news:20070411205846.966$k1...@newsreader.com...
[snip]

> > > > plot(x^12-2^x,x=-infinity..infinity);
>
> Very interesting. Mathematica, for example, balks at any request to graph
> from -Infinity to Infinity.

[snip]

> This probably deserves a new thread, so I've changed the title
> to "Comprehensive Graphing".
>
> I've used such graphs for many years (although I did not produce such a
> graph for x^12 - 2^x until after you had posted the graphs produced by
> Maple). I have called them "comprehensive graphs". Is there an "accepted"
> name for them?
>
> For comparison with your Maple 9 graph, here's a comprehensive graph which
> I produced with Mathematica:
>
> <http://img264.imageshack.us/img264/4936/comprehensivegraphmq6.gif>
>
> The code I used was
>
> f[x_] := x/(1 - Abs[x]); fInv[x_] := x/(1 + Abs[x]); g[x_] := x^12 - 2^x;
> Plot[fInv[g[f[x]]], {x, -1, 1}, AspectRatio -> Automatic, Ticks -> False]

I think you just answered one of the questions posted back in 2003:

http://tinyurl.com/2sw7ye

Your solution seems to be a reasonable one. I am going to examine it closer
tomorrow (cause I need to catch some z's now, it's 4:20am), but it seems that
you are exhibiting a mapping very similar to the one Maple uses for such
graphs. Nice.

> Above, you can see the scaling function f which I used. It's my "default"
> for comprehensive graphing, but of course many other such functions could
> be used (e.g. Tanh for f, ArcTanh for fInv). What scaling function does
> Maple use by default for such graphs? Can the user specify alternative
> scaling functions?

Maybe if someone posted the same question over to comp.soft-sys.math.maple,
someone there will know. It is clear to me however that Maple applies some
transform which maps both axes (-oo, +oo) to [a,b] (or (a,b)), with a < 0 < b,
and a, b finite.

> > The Maple 9 graph also seems to be wrong, although "less wrong" than the
> > Maple V graph. I didn't check this one when I answered the question, but
> > to me it implies that the function reaches +infinity between the second
> > and third root,
>
> It doesn't imply that to me. Rather, it just shows that the function gets
> very large between the second and third roots.

I would agree, if there was some positive distance between +infinity (on top)
and the max of the function between the second and third root. As it stands,
that distance looks like 0, which hints that +infinity is actually reached,
especially since Maple 9 labels the top/right as +infinity.

Even though it's really a minor point, it can be really misleading for someone
looking over at such graphs. In particular, if f(x) = x^12 - 2^x, the maximum
between the second and third zeroes seems to occur at:

x_c = -11*W(-1,-1/132*12^(10/11)*ln(2)^(12/11))/ln(2) (the dreaded W again...)

~= 71.97806937 (unless I misread the zeros of df/dx again)

and f(x_c) ~= .1468646499e23.

That's indeed huge. But it's still not infinity. *Some* indication should be
given in my opinion by the graph to differentiate it from infinity under the
"comprehensive" mapping, although I understand your point below, that if you
also map the y-axis to something like [-1,1], indeed f(x_c) will be mapped
very close to 1. I suspect that under such a map, the difference will be so
tiny that the plotting algorithm is not able to differentiate between f(x_c)
and +infinity on the y-axis.

> > and that the third root is very large in magnitude,
> > almost close to +infinity.
>
> Well, that's presumably because it is! In my comprehensive graph, for
> example, the distance between 0 and +oo is 1, while the distance between
> that third root and +oo is merely 0.013...

Agreed. I will grant you that. It's probably the nature of the transform, so
perhaps this can be excused. Note however that even though small, the
difference between the third root and +infinity still shows. On the y-axis,
the difference between f(x_c) and +infinity doesn't.

> David

David W. Cantrell

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Apr 12, 2007, 1:20:39 AM4/12/07
to

But I'm sure you'd realize, if you thought about it a little, that one
can _never_ take for granted that a function takes on a specific value
(be it 0, 1, +oo, or any other) merely because a computer-generated graph
makes it _look_ as though it does. (There are only so many pixels, etc.)
One must always adopt the attitude that the graph simply makes it look as
though the function _might_ take on that specific value.

> In particular, if f(x) = x^12 - 2^x,
> the maximum between the second and third zeroes seems to occur at:
>
> x_c = -11*W(-1,-1/132*12^(10/11)*ln(2)^(12/11))/ln(2) (the dreaded W
> again...)
>
> ~= 71.97806937 (unless I misread the zeros of df/dx again)
>
> and f(x_c) ~= .1468646499e23.
>
> That's indeed huge. But it's still not infinity. *Some* indication should
> be given in my opinion by the graph to differentiate it from infinity
> under the "comprehensive" mapping, although I understand your point
> below, that if you also map the y-axis to something like [-1,1], indeed
> f(x_c) will be mapped very close to 1. I suspect that under such a map,
> the difference will be so tiny that the plotting algorithm is not able to
> differentiate between f(x_c) and +infinity on the y-axis.

Right. However, if you really insist on _seeing_ in a comprehensive graph
that the function is not +oo between the second and third roots, that can
certainly be done. Just use some different, appropriate scaling function.
One such scaling function, which I've used before, is

f[x_] := Sign[x](Exp[Abs[x]/(1 - Abs[x])] - 1)

with its inverse

fInv[x_] := Sign[x] Log[1 + Abs[x]]/(1 + Log[1 + Abs[x]])

David

Ioannis

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Apr 12, 2007, 12:12:07 PM4/12/07
to
I, "Ioannis" <morp...@olympus.mons> wrote in message
news:1176342738.558197@athprx03...

I played with your transform with Maple today, as follows:

>f:=x->x/(1 - abs(x));
>fInv :=x-> x/(1 + abs(x));
>g :=x-> x^12 - 2^x;
>with(plots):
>plot(fInv(g(f(x))), x=-1..1);

Here are the results:

Maple V release 4:
http://misc.virtualcomposer2000.com/graph3.gif

Maple 9:
http://misc.virtualcomposer2000.com/graph4.gif

It appears as though Maple V still misbehaves. Curiously, if I do a:

>limit(fInv(g(f(x))),x=1,left);

Maple V outputs correctly: -1, so I fail to see why it doesn't plot the
transform right.

I think your transform graph (and graph3.gif which is still erroneous)
conspires highly towards the fact that Maple's transform (at least Maple 9's)
must be very similar to yours, but I cannot understand why Maple V doesn't
graph it right close to x=1.

> > Above, you can see the scaling function f which I used. It's my "default"
> > for comprehensive graphing, but of course many other such functions could
> > be used (e.g. Tanh for f, ArcTanh for fInv). What scaling function does
> > Maple use by default for such graphs? Can the user specify alternative
> > scaling functions?

gyr...@gmail.com

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Aug 17, 2014, 8:01:30 AM8/17/14
to
On Tuesday, April 10, 2007 1:24:00 PM UTC-7, David Pekker wrote:
> how many roots does this equation have?
>
> Thanks in advance for explaining

. W O N .

Peter Percival

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Aug 17, 2014, 9:02:44 AM8/17/14
to
gyr...@gmail.com wrote:
> On Tuesday, April 10, 2007 1:24:00 PM UTC-7, David Pekker wrote:
>> how many roots does this equation [x^12 = 2^x] have?

Infinitely may complex ones but only three real ones.

For x around -1 to +1 both x^12 and 2^x are rather flat and there will
be roots near -1 and +1; but eventually 2^x grows much faster than x^12
so there is another root for sufficiently big x (maybe x = 74 or 75).
Analytic solutions (real and complex) use the Lambert W function or its
relatives.

--
[Dancing is] a perpendicular expression of a horizontal desire.
G.B. Shaw quoted in /New Statesman/, 23 March 1962

gyr...@gmail.com

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Aug 17, 2014, 10:54:17 AM8/17/14
to
sci.math ›
roots of x^12 = 2^x

KNOW YOUR’ SPEA’ KING GREK!

X!@@X = 3.464101615137754
X!@@X = X^II=II^X
X!@@X = X^IIIII^X
X^I | I | = 1 | 1^X
x^12 |x| 2^x
x^12 |=| 2^x
x^12 |^| 2^x
X^!@|X|@^X
X^!@|+|@^X
X^!@|^|@^X
x^12 = x^2

X^12=2^X
X^XII=II^X
X^II=X^II
x^12 = x^2
X^!@ + X^@
x^12 = 2x^
^x2 = 21^x
x^12 = x^II
X^12 = 2^X
X^XII = II^
X^II = X^II
^XII = X^II
^XII = 12^X
II^X = X^2
x12^=xII^

#HANG GLIDING OVER THE CPAST OF NEPAL! ( HG OTC ON )

2^3.464101615137754 = 12^1.414213562373095
23.464101615137754 = 121.414213562373095

11.11144012336555
3.333382684806164
1.825755373757986
1.825755373757986
1.162413504764159
1.078152820691093
1.038341379648857
1.018990372696846
1.009450530088942
1.004714153423222
1.002354305334806
1.001176460637587
1.000588057413033
1.000293985492781
1.000146981944545
1.000073488272009
1.000036743460964
1.000018371561725
1.000009185738673
1.00000459285879
1.000002296426758
1.00000114821272
1.000000574106195
1.000000287053056
1.000000143526518
1.000000071763256
1.000000035881627
1.000000017940814
1.000000008970407
1.000000004485203
1.000000002242602
1.000000001121301
1.00000000056065
1.000000000280325
1.000000000140163
1.000000000070081
1.000000000035041
1.00000000001752
1.00000000000876
1.00000000000438
1.00000000000219

1.000000000001095
1.000000000000547
1.000000000000274
1.000000000000137
1.000000000000068
1.000000000000034
1.000000000000017
1.000000000000008
1.000000000000004
1.000000000000002
1.000000000000001
1
1
1

#

gyr...@gmail.com

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Aug 17, 2014, 11:27:38 AM8/17/14
to
On Tuesday, April 10, 2007 1:24:00 PM UTC-7, David Pekker wrote:
> how many roots does this equation have?
>
> Thanks in advance for explaining

!KNOW YOUR' SPEA' KING GREK!

Q: The roots of x^12 = 2^x?

A: |^|San Hose California.

example:

gyr...@gmail.com

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Aug 17, 2014, 1:00:35 PM8/17/14
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On Tuesday, April 10, 2007 1:24:00 PM UTC-7, David Pekker wrote:
> how many roots does this equation have?
>
> Thanks in advance for explaining


x^12 = 2^x

X!@ @@X !X!

X@!

!@X
@
x^12 = 2^x
X@!=!@X=@X!
@!=!@X=@X!X
!=!@X=@X!X@
=!@X=@X!X@!
!@X=@X!X@!=
@X=@X!X@!=!
X=@X!X@!=!@
=@X!X@!=!@X
@X!X@!=!@X=
X!X@!=!@X=@
!X@!=!@X=@X
X@!=!@X=@X!
2^x = 12^x
@
@X!


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