Question

Solve the following system of equations.
$2({\log }_{y}x+{\log }_{x}y)=5,xy=8.$

Answer 1

$2({\log }_{y}x+{\log }_{x}y)=5$

$2\left[\frac{{\left(\log x\right)}^2+{\left(\log \left(8/x\right)\right)}^2}{\left(\log x\right)\left(\log \left(8/x\right)\right)}\right]=5\to 1$

$xy=8$

$xy=8\to 2$

From $1$ and $2$

$2\frac{[{\left(\log x\right)}^2+{(\log 8-\log x)}^2]}{\left(\log x\right)\left(\log \left(8/x\right)\right)}=5$

$2{\left(\log x\right)}^2+2{\left(\log 8\right)}^2+2{\left(\log x\right)}^2-4\left(\log 8\right)\left(\log x\right)=5\left(\log x\right)\left(\log 8\right)-5{\left(\log x\right)}^2$

$9{\left(\log x\right)}^2-9\left(\log 8\right)\left(\log x\right)+2{\left(\log 8\right)}^2=0$

$[3\log x-\left(\log 8\right)][3\log x-\left(\log 8\right)]=0$

$\log x=\frac{2}{3}\left(\log 8\right)\log x=\frac{1}{3}\log 8$

$x=4,2;y=2,4$

$(x,y):(4,2),(2,4)$