Question

rezolvati in multimea numerelor reale ecuatia ㏒₂(x²+4)=㏒₂8

Answer 1

[tex]log_2(x^2+4)=log_28 \rightarrow x^2+4=8\\ x^2=4\\ x=\pm2[/tex]

Answer 2

$\displaystyle log_2\left(x^2+4 \right)=log_28~~~~~~~~~~~~~~~~~~~~~~~~~C.E.~x^2+4\ \textgreater \ 0\\ \\~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\boxed{log_b(f(x))=log_b(g(x)) \Rightarrow f(x)=g(x)} \\ \\ ~x^2+4=8 \Rightarrow x^2=8-4 \Rightarrow x^2=4 \Rightarrow x= \sqrt{4} \Rightarrow x_1=2 \\ \\~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\Rightarrow x_2=-2$

$\displaystyle x_1=2 \Rightarrow 2^2+4\ \textgreater \ 0~A \Rightarrow x_1=2 ~este~solutie~a~ecuatiei \\ \\ x_2=-2 \Rightarrow (-2)^2+4\ \textgreater \ 0~A \Rightarrow x_2=-2~este~solutie~a~ecuatiei \\ \\ S=\{-2,2\}$