Question

2log in baza 2 din X - log in baza X din 2 = 1 . A se rezolva in multimea nr reale

Answer 1

$\displaystyle\bf\\2log_2\,x-log_x\,2 = 1\\\\\text{folosim formula: }~~log_a\,b=\frac{1}{log_b\,a}\\\\2log_2\,x-\frac{1}{log_2\,x}-1=0\\\\\text{Facem substitutia: }~~\boxed{\bf~log_2\,x=t}\\\\2t-\frac{1}{t}-1=0~\Big|\cdot t\\\\2t^2-1-t=0\\\\2t^2-t-1=0$  

.

$\displaystyle\bf\\t_{12}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}=\frac{1\pm\sqrt{1+8}}{4}=\frac{1\pm\sqrt{9}}{4}=\frac{1\pm3}{4}\\\\t_1=\frac{1+3}{4}=\frac{4}{4}=1~\implies~log_2\,x=1~\implies~\boxed{\bf~x_1=2}\\\\t_2=\frac{1-3}{4}=\frac{-2}{4}=-\frac{1}{2}\\\\\implies~log_2\,x=-\frac{1}{2}~\implies~x_2=\boxed{\bf2^{^{-\dfrac{1}{2}}}}=\frac{1}{2^{\frac{1}{2}}}=\frac{1}{\sqrt{2}}=\boxed{\bf\frac{\sqrt{2}}{2}}$