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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 21,812

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

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 21,812

Mathsy, I think you missed this post!

With all the available technology, you could have well improved upon those digits!

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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

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

-----------------------------------------

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

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

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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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,631

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

thanks MathsIsFun! : )

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 21,812

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 + ∞

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 21,812

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!

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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

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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