Question

Rezolvați în mulțimea nr reale ecuația: log în baza2 x+1/log în baza 2x=2​

Answer 1

 

$\displaystyle\bf\\\frac{log_2(x+1)}{log_2(x)}=2\\\\\textbf{Folosim formula:}~~~\frac{log_cb}{log_ca}=log_ab\\\\\\\implies~~~\frac{log_2(x+1)}{log_2(x)}=log_x(x+1)\\\\Rescriem~ecuatia:\\\\log_x(x+1)=2\\\\x^2=x+1\\\\x^2-x-1=0\\\\x_{12}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\x_{12}=\frac{1\pm\sqrt{1+4}}{2}\\\\x_{12}=\frac{1\pm\sqrt{5}}{2}\\\\\boxed{\bf~x_1=\frac{1-\sqrt{5}}{2}}\\\\\boxed{\bf~x_2=\frac{1+\sqrt{5}}{2}}$