Question

rezolvati ecuatia log in baza 2 din x = log in baza 4 din (3x-2)

Answer 1

$\displaystyle log_2x=log_4(3x-2)~~~~~~~~~~~~~~~~C.E.~ \left \{ {{x\ \textgreater \ 0} \atop {3x-2\ \textgreater \ 0}} \right. \Rightarrow \left \{ {{x\ \textgreater \ 0} \atop {x\ \textgreater \ \frac{2}{3} }} \right. \\ \\ log_2x= \frac{log_4x}{log_42} \\ \\ \frac{log_4x}{log_42} =log_4(3x-2) ~~~~~~~~~~~~~~~~~~~log_42=log_4 \sqrt{4} = \frac{1}{2} log_44= \frac{1}{2} \\ \\ \frac{log_4x}{ \frac{1}{2} } =log_4(3x-2) \\ \\ 2log_4x=log_4(3x-2) \\ \\ log_4x^2=log_4(3x-2) \\ \\ x^2=3x-2 \\ \\ x^2-3x+2=0 \\ \\ a=1,~b=-3,~c=2$

$\displaystyle \Delta=b^2-4ac=(-3)^2-4 \cdot 1 \cdot 2=9-8=1\ \textgreater \ 0 \\ \\ x_1= \frac{-b+ \sqrt{\Delta} }{2a} =\frac{3+ \sqrt{1} }{2 \cdot 1} = \frac{3+1}{2} = \frac{4}{2} =\boxed2 \\ \\ x_2=\frac{-b- \sqrt{\Delta} }{2a} = \frac{3- \sqrt{1} }{2 \cdot 1} = \frac{3-1}{2} = \frac{2}{2} =\boxed1$