Question

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 ?

Answer 1

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