My first approach involved simply taking the product: $$xy=a^bb^a=b^{b\log_b(a)+a}$$ However, I can't really do anything to, for instance, separate the exponent from the base $b$. Additionally, division has basically the same problem, and addition and subtraction are quite messy. Therefore, I settled on exponentiation and logarithms for my second and third approaches. $$x^y=({a^b})^{b^a}=a^{b^{a+1}}$$ $$\log_x(y)=\frac{\ln(y)}{\ln(x)}=\frac{a\ln(b)}{b\ln(a)}=\frac{a}{b}\log_a(b)$$ The third approach looked especially promising in conjunction with the first approach, but there was no way to surpass the $b^\mathrm{<exponent>}$ barrier.
EDIT
I am looking for algebraic solution, that can apply to real numbers, instead of number theory solutions, that can only apply to the integers. Thank you.