Question

Rezolvati in R ecuatia log3(2x+1)-log3(2x-1)= -1

Answer 1

Răspuns:

Explicație pas cu pas:

$MVA:~\left \{ {{2x+1>0} \atop {2x-1>0}} \right. ~\left \{ {{x>-1/2} \atop {x>1/2}}~deci~x>1/2. \right. \\log_{3}(2x+1)+1=log_{3}(2x-1),~log_{3}(2x+1)+log_{3}(3)=log_{3}(2x-1),~log_{3}(3(2x+1))=log_{3}(2x-1),~3(2x+1)=2x-1,~6x+3=2x-1,~6x-2x=-1-3,~4x=-4,~x=-1.$

-1∉MVA, deci ecuaţia nu are soluţii

Answer 2

$\log_{3}(2x+1) - \log_{3}(2x-1) = -1\\\\\Rightarrow \log_{3}\dfrac{2x+1}{2x-1} = -1\\ \\ \Rightarrow \log_{3}\dfrac{2x+1}{2x-1} = \log_{3}\dfrac{1}{3}\\ \\ \Rightarrow \dfrac{2x+1}{2x-1} = \dfrac{1}{3} \\ \\ \Rightarrow 3(2x+1) = 2x-1\\ \\ \Rightarrow 6x+3 = 2x-1\\ \\ \Rightarrow 4x = -4\\ \\ \Rightarrow x = -1\\ \\\\ \text{Verificare:}\quad \log_{3}1-\log_{3}(-3)= -1\quad (F)\\ \\ \Rightarrow \boxed{S = \emptyset}$