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?
https://www.desmos.com/calculator/ly4es9gpq2
The first equation, in red, shows two distinct branches. One with x<y and one with x>y. This implies either a or b could be larger.
It is also interesting to note that for any value of a>1 there appear to be two values of b. For example, if a=3, b can be either 2 or about 3.221.
The same is true for b, except it cannot be between about 2.018 and 3.196. Also if b is in the lower range both a's will be larger and visa versa.
The second equation, in green, shows all solutions for x^y=y^x with x=1 and y=1 as asymptotes. It can also be that x=y, but this is partially obscured by the third equation.
The third equation is x=y, a blue dotted line separate the regions for the first question.
Edited on September 8, 2018, 5:14 pm

Posted by Jer
on 20180908 17:06:13 