Two positive distinct numbers fulfill the equation
b^{a} a^{b}=1
Which is larger:a or b?
For what (x,y) couple are both terms i.e x^{y} and y^{x} equal?
For part 1, involving a and b, it's impossible to tell whether a or b is larger. Take for example
In 4^1.8871211878830906589...  1.8871211878830906589...^4 = 1, b>a
In 2.0940128528581167022... ^4  4^2.0940128528581167022... = 1 a>b
Part 2:
x^y = y^x
y*log(x)=x*log(y)
y/x = log(y)/log(x)
y/log(y) = x/log(x)
This can be solved for a value for the two sides is above e, when, i.e., when x/log(x) has two values.
For example, when Wolfram Alpha is asked to solve x/log(x) = 3, using natural logs, the two exact answers it gives are e^(W(1/3)) and e^W{1}(1/3) where {} indicates a subscript, where W is the productlog or Lambert Wfunction. With a 1 subscript I'm not sure what it means. But 1.8571838602078353365 and 4.536403654973527422 are numerical approximations to x/log(x)=3, so either one raised to the other's power will be the same value, 16.5819024004183....
In fact, 2^4 = 4^2, an integral solution near the above real solution.

Posted by Charlie
on 20180908 13:23:42 