All possible solutions to $x^y = 8$ and $y^x = 9$

Question

I know the obvious solution $(x,y)=(2,3)$. I just want to know if there are any other solutions to above system of equations or is $(x,y)=(2,3)$ the only solution. Also, both $x$ and $y$ should be real numbers.

Please provide a method to solve equations like these if it exists.

Answer 1

I wrote it as a comment but i will complete it now ,

well, you have two equation with two variables and because the $\ln(x)$ and $e^x$ is one-to-one functions in the Real domain there is one solution. if you allow $x,y$ to be complex numbers then because $\ln(x),e^x$ are periodic function in the complex domain,i think there could by more than one answer.(actually infinitely many)

for instance : take $x=1+i$ then for all $y=\frac{i \left(2 \pi c-3 i \log (2)\right)}{\log (1+i)}$ for any integer $c$.

hope its what you are looking for.

Answer 2

if you take the logarithm on both sides we get $$y\ln(x)=\ln(8)$$ and $$x\ln(y)=\ln(9)$$ from here we get $$y\ln\left(\frac{\ln(9)}{\ln(y)}\right)=\ln(8)$$ and here you must use a numerical method.