x^log(y)+y^log(x^(1/2))=110

xy=1000

This is my attempt:

log(y)*log (x)+log (x^(1/2)*log(y)=log 110

y=1000/x

log(1000/x)*log (x)+log (x^(1/2)*log(1000/x)=log 110

log (1000)-log (x)*log (x)+log (x^(1/2)*log(1000)-log(x)=log 110

log(x)=t

log (1000)=3

(3-t)t+(t/2)(3-t)=log 110

3t^-3t^2+(3t-t^2)/2=log 110

I believe this is an incorrect solution, can you please tell where my error is? If my attempt is incorrect from the beginning, please show me how to solve it. Thank you in advance!