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
]]>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!
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 + ∞
]]>I will let Ganesh reply to this, but just thought I would say hi.
]]>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.
]]>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
]]>