
Re: x^y = y^x What number meaning?
Posted:
Aug 14, 2017 11:18 PM


On Sun, 13 Aug 2017 21:32:35 0500, Serg io wrote:
>On 8/13/2017 6:04 PM, Jim Burns wrote: >> On 8/13/2017 4:43 PM, Peter Percival wrote: >>> Mr Sawat Layuheem wrote: >> >>>> x^y = y^x What number meaning? >>> >>> The solution is >>> y = x W(log(x)/x)/log(x) >>> where W is Lambert's W function. See >>> https://en.wikipedia.org/wiki/Lambert_W_function. >> >> Also, >> x^y = y^x >> >> x^(y/x) = y >> >> x^(y/x  1) = y/x >> >> Let y = m*x >> >> x^(m1) = m >> >> x = m^(1/(m1)) >> >> y = m*m^(1/(m1)) = m^(m/(m1)) >> >> That gives us lots of examples (x,y), one for each m. >> >>  >> Hmmm. A slightly prettier answer: >> >> k = 1/(m1) >> >> m = (k+1)/k >> >> x = ((k+1)/k)^k >> >> y = ((k+1)/k)^(k+1) >> >> k x y >> 1 2 4 >> 2 9/4 27/8 >> 3 64/27 256/81 >> ... >> > >that is a nice solution
Indeed.
Keeping some parts separated for speed in a computer program (only one exponentiation needed):
m = (k + 1) / k x = m^k
y = x * k
But it loses the beauty of the fractions in Jim's post.

