Question

If $lo{g}_{2}x+lo{g}_{x}2=\frac{10}{3}=lo{g}_{2}y+lo{g}_{y}2$ and $x\ne y$
then prove that x + y = 8 + 2${}^{1/3}$

Answer 1

$\frac{10}{3}=3+\frac{1}{3}$
The given equation is of the form of
$p+\frac{1}{p}=3+\frac{1}{3}=q+\frac{1}{q}$
Where $p\ne qasx\ne y$
$\therefore lo{g}_{2}x=3,lo{g}_{2}y=1/3$
$\therefore x=2^3,y=2^{1/3}$
$\therefore x+y=8+2^{1/3}$