Question

# TUTUR TINUE a (D) a prime numb Find all.real numbers x which satisfy the equation 2 log, log2 X + log1/2 log2 (212 x)= 1.

Solution

( quad 2 log _{2} log _{2} x+log _{1 / 2} log _{2}(2 sqrt{2} x)=1 )

( Rightarrow 2 log _{2} log _{2} x=log _{2} log _{2}(2 sqrt{2} x)=1 )

( Rightarrow log _{2}left(frac{left.log _{2} xright)^{2}}{log _{2}(2 sqrt{2} x)}=1right. )

( Rightarrow quadleft(log _{2} xright)^{2}=2 log _{2}(2 sqrt{2} x) )

( Rightarrowleft(log _{2} xright)^{2}=log _{2}left(8 x^{2}right) )

( Rightarrowleft(log _{2} xright)^{2}=log _{2} 8+2 log _{2} x )

( Rightarrowleft(log _{2} xright)^{2}-2 log _{2} x-3=0 )

( Rightarrowleft(log _{2} x-3right)left(log _{2} x-1right)=0 )

( Rightarrow log _{2} x=1,3 )

( x=2^{prime}, 2^{3} )

( x=2,8 )