Question

Let $y=\sqrt{{\log }_{2}3\cdot {\log }_{2}12\cdot {\log }_{2}48\cdot {\log }_{2}192+16}-{\log }_{2}12\cdot {\log }_{2}48+10$. Find $y\in N$.

Answer 1

Answer: 6

$y=\sqrt{{\log }_{2}3\cdot \left(2+{\log }_{2}3\right)\cdot \left(4+{\log }_{2}3\right)\cdot \left(6+{\log }_{2}3\right)+16}-\left(2+{\log }_{2}3\right)\cdot \left(4+{\log }_{2}3\right)+10$
let ${\log }_{2}3=x$
$y=\sqrt{x(2+x)(4+x)(6+x)+16}-(2+x)\cdot (4+x)+10$
$=\sqrt{\left(x^2+6x\right)\left(x^2+6x+8\right)+16}-\left(x^2+6x+8\right)+10$
let $x^2+6x=t$
$y=\sqrt{t(t+8)+16}-(t+8)+10$
$=\sqrt{t^2+8t+16}-(t-2)$
$=t+4-t+2=6$
hence $y=6$