اگر $x=\log (۲\sqrt{۲}-۱)+\log (\sqrt{۲}+۱)$ و $y={{۱۰}^{x}}-۳$ باشد، حاصل ${{\log }_{۲}}y$ کدام است؟
${{10}^{x}}={{10}^{\log (2\sqrt{2}-1)}}\times {{10}^{\log (\sqrt{2}+1)}}=(2\sqrt{2}-1)(\sqrt{2}+1)\Rightarrow {{10}^{x}}=4+2\sqrt{2}-\sqrt{2}-1=3+\sqrt{2}\Rightarrow y={{10}^{x}}-3=\sqrt{2}$ ${{\log }_{2}}y={{\log }_{2}}\sqrt{2}=\frac{1}{2}$