## anonymous 5 years ago If x,y>0, log base y(x)+log base x(y)=10/3 and xy=144, what is the value of (x+y)/2 ?

$\log_y{x}+\log_x{y}=10/3 , log_y{x}=log^{-1}_x{y}=z$ $z+z^{-1}=10/3$ $z^2-10z/3+1=0$ $z=3\,\,or\,\,z=1/3$ $x=y^3\,\,or\,\,y=x^3$ $(x,y)=\{(24\sqrt{3},2\sqrt{3}),(2\sqrt{3},24\sqrt{3})\}$ $\frac{x+y}{2}=13\sqrt{3}$