Question

Solve the following system of equations.
$x^{\log y}=2,xy=20$

Answer 1

$x^{\log y}=2xy=20$

$x^{\log y}=2\to 1xy=20\to 2$

Taking log on both sides

$\left(\log x\right)\left(\log y\right)=\log 2$

From $2$

$\left(\log x\right)\log \left(\frac{20}{x}\right)=\log 2$

$\log x(\log 20-\log x)=\log 2$

${\left(\log x\right)}^2-\left(\log 20\right)\left(\log x\right)+\log 2=0$

$\log x=\frac{+\left(\log 20\right)\pm \sqrt{{\left(\log 20\right)}^2-4\log 2}}{2}$

$\log x=\frac{\left(\log 20\right)\pm \sqrt{{(2\log 2+\log 5)}^2-4\log 2}}{2}$

$\log x=\frac{\left(\log 20\right)\pm \sqrt{{4\left(\log 2\right)}^2+{\left(\log 5\right)}^2+4\log 2-4\log 2}}{2}$

$\log x=\frac{\left(\log 20\right)\pm \sqrt{{4\left(\log 2\right)}^2+{\left(\log 5\right)}^2+4\log 2(\log \left(5\right)-\log 10)}}{2}$

$\log x=\frac{\left(\log 20\right)\pm \sqrt{{4\left(\log 2\right)}^2+{\left(\log 5\right)}^2+4{\left(\log 3\right)}^2}}{2}$

$\log x=\frac{(\log 20\pm \log 5)}{2}$

$\log x=\frac{(2\log 2+\log 5\pm \log 5)}{2}$

$\log x=\log 2+\log 5,\log 2$

$\log x=\log \left(10\right),\log 2$

$x=10,2$

From $2$ $xy=20$

$y=\frac{20}{x}y=\frac{20}{10}=2,y=\frac{20}{10}=10$

$(x,y):(10,2),(2,10)$