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#1 2005-07-13 21:39:52

ganesh
Moderator
Registered: 2005-06-28
Posts: 15,159

Which of the two is greater?

When there are two numbers x and y, such that both x,y ≥1,
does it follow that y^x is always greater than x^y if x is greater than y?
No.
This is true only if y is greater than a certain CRITICAL Value.
Many years back, I tried to find this critical value of y for certain values of x.

Value of x                             Approximate value of y

10                                        1.3712886
100                                      1.04955
1000                                     1.0069805
10,000                                  1.000922309
100,000                                1.00011514925
1,000,000                             1.0000138158
10,000,000                           1.00000161283
100,000,000                         1.0000001843
1,000,000,000                      1.0000000208

Illustration:- y^100 can be greater than 100^y only if the value of y ≥1.04955


Character is who you are when no one is looking.

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#2 2005-07-22 22:25:51

ganesh
Moderator
Registered: 2005-06-28
Posts: 15,159

Re: Which of the two is greater?

Mathsy, I think you missed this post!
With all the available technology, you could have well improved upon those digits! smile


Character is who you are when no one is looking.

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#3 2005-07-22 23:39:03

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Which of the two is greater?

Occasionally, the box goes away before I can read all the new posts. Usually when there have been lots of new posts and it takes me a long time to read them all. I think that's what happened here.

There's a strong pattern emerging there, though.
Do you think it's possible to rearrange x^y=y^x to find y in terms of x?
If you could do that, you could find the critical value for any value of x.

I've got as far as the yth root of y = the xth root of x, but now I'm stuck.


Why did the vector cross the road?
It wanted to be normal.

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#4 2005-07-24 13:01:52

kylekatarn
Member
Registered: 2005-07-24
Posts: 445

Re: Which of the two is greater?

-----------------------------------------
I think that finding these crit. pts. should involve logarythms. just a supposition=P

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#5 2005-07-24 14:22:22

kylekatarn
Member
Registered: 2005-07-24
Posts: 445

Re: Which of the two is greater?

this topic is amazing!

I did some 'LN' transformations to the expression and I found the equation:

ln( y^(1/y) ) = ln(x)/x

Solving this equation in a computer program like mathematica, maple or derive, you can find the critical value yCrit; for a given x
for example, I was able to compute yCritical for several  x's with 50 precision digits(but this can be increased):
x = xValue            y = yCritical
-------------------------------------------------------------------------------------------------
x=10                    y = 1.3712885742386235368613621062996899588428544048422
x=100                  y = 1.0495191898071712311474936519440559096925868204045
x=1000                y = 1.0069802219160264731969790392939479509214698343986
x=10000              y = 1.0009223085800102005258019267508413188152496261875
x=100000            y = 1.0001151491408378890243699386042389677717403925799
x=1000000          y = 1.0000138157968674942789013367960898614105090318224
x=10000000        y = 1.0000016118134620021095317020510233099984034273812
x=100000000      y = 1.0000001842068583377621834070851145767978026108316
x=1000000000    y = 1.0000000207232664811270553117793424621783832233279
x=10000000000  y = 1.0000000023025851009468928822904460936293772528616
-------------------------------------------------------------------------------------------------
it seems that yCritical "slowly"(?) approaches the limit of 1 as x approaches +oo

looking forward to hear comments on this topic!

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#6 2005-07-24 18:08:09

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,552

Re: Which of the two is greater?

Hello, and welcome to the forum kylekatarn !

I will let Ganesh reply to this, but just thought I would say hi.


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#7 2005-07-25 01:23:56

kylekatarn
Member
Registered: 2005-07-24
Posts: 445

Re: Which of the two is greater?

thanks MathsIsFun! : )

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#8 2005-07-25 16:20:06

ganesh
Moderator
Registered: 2005-06-28
Posts: 15,159

Re: Which of the two is greater?

kylekatarn wrote:

it seems that yCritical "slowly"(?) approaches the limit of 1 as x approaches +oo

looking forward to hear comments on this topic!

Yes, you are correct! ycritical approaches 1, but is certainly greater than 1, as x approaches + ∞


Character is who you are when no one is looking.

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#9 2005-07-31 22:21:46

ganesh
Moderator
Registered: 2005-06-28
Posts: 15,159

Re: Which of the two is greater?

mathsyperson wrote:

I've got as far as the yth root of y = the xth root of x, but now I'm stuck.

Interestingly, if yth root of y = xth root of x,
it does not automatically follow that x=y
For example, if x=4 and y=2,
then this is true! roll


Character is who you are when no one is looking.

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#10 2005-08-07 11:05:54

NIH
Member
Registered: 2005-06-14
Posts: 33

Re: Which of the two is greater?

It's not possible to express y explicitly in terms of x using any of the standard elementary functions.  However, there is a formal solution using something called the Lambert W function.  This function can be evaluated using mathematical packages such as Maple and Mathematica, but no calculator currently has a button for it.  See the references below for details.

http://mathworld.wolfram.com/LambertW-Function.html
http://www.americanscientist.org/template/AssetDetail/assetid/40804;_voi8-8bIm
http://www.orcca.on.ca/LambertW/

Another approach would be to use the Newton Raphson method.  If a is an approximation to a root of
f(x) = 10^x - x^10 = 0, then a - f(a)/f'(a) will be a better approximation.
In this case, we have f'(x) = ln(10) * 10^x - 10x^9.

For example, if a = 1.4 is an approximate solution, then
1.4 - (10^1.4 - 1.4^10)/(ln(10) * 10^1.4 - 10*1.4^9) ~= 1.3744 is a better approximation.

This converges quite rapidly.  The next two convergents are, to 10 decimal places, 1.3713296532 and
1.3712885814.

http://www.sosmath.com/calculus/diff/der07/der07.html

Finally, here's a page which will solve, for example, the equation 10^x = x^10, giving several answers in terms of the Lambert W function (aka the ProductLog function), along with the numeric values.

http://www.hostsrv.com/webmab/app1/MSP/quickmath/02/pageGenerate?site=quickmath&s1=equations&s2=solve&s3=basic


2 + 2 = 5, for large values of 2.

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