Question

sa se rezolve in multimea nr. reale
ecuatia log7 (6x-12)=log7(x la a2 -4)

Answer 1

$\displaystyle log_7(6x-12)=log_7(x^2-4) \\ \left \{ {{6x-12\ \textgreater \ 0} \atop {x^2-4\ \textgreater \ 0}} \right.\Rightarrow \left \{ {{x\ \textgreater \ 2} \atop {x\ \textgreater \ 2,~x\ \textless \ -2}} \right. \\ \left\begin{array}{ccc}x \in (2,\infty)\\x\in (-\infty,-2) \cup(2,\infty)\\&&\end{array}\right\}\Rightarrow x \in (2,\infty)$
$\displaystyle \\ 6x-12=x^2-4 \\ -x^2+6x=-4+12 \\ -x^2+6x=8 \\ -x^2+6x-8=0~| \cdot (-1) \\ x^2-6x+8=0$
$\displaystyle \Delta=(-6)^2-4 \cdot 1 \cdot 8=36- 32=4\ \textgreater \ 0 \\ x_1= \frac{6+ \sqrt{4} }{2 \cdot 1} = \frac{6+2}{2}= \frac{8}{2} =4 \\ x_2= \frac{6- \sqrt{4} }{2 \cdot 1} = \frac{6-2}{2} = \frac{4}{2} =2 \\ \boxed{x=4}$