Question

Solve the following system of equations.
$5{\log }_{2}x={\log }_2{y}^3-{\log }_{\sqrt{2}}2,{\log }_{2}y=8-{\log }_{\sqrt{2}}x$

Answer 1

$5{\log }_{2}x={\log }_2{y}^3-{\log }_{\sqrt{2}}2$

${\log }_2{x}^5={\log }_2{y}^3-{\log }_{2}2$

${\log }_2{x}^5={\log }_2\left(\frac{y^3}{4}\right)$

$4x^5=y^3\to 1$

${\log }_{2}y=8-{\log }_{\sqrt{2}}x$

${\log }_{2}y=8-2{\log }_{2}x$

${\log }_{2}y+2{\log }_{2}x=8$

${\log }_2\left(y{x}^2\right)=8$

$x^{2}y=2^8$

${\left(x^{2}y\right)}^3={\left(2^8\right)}^3$

$x^6{y}^3=2^{2^4}$

$y^3=\frac{2^{2^4}}{x^6}\to 2$

From $1$ and $2$

$4x^5=\frac{2^{2^4}}{x^6}$

$x^{11}={\left(2\right)}^{22}$

$x={\left(2\right)}^2=4$

From $1$: $y^3=4x^5=4{\left(4\right)}^5={\left(4\right)}^6$

$y={\left(4\right)}^2=16$

$(x,y):(4,16)$