Question

$3*4^{x} - 6^{x} = 2* 9^{x}$ ????

Answer 1

Se imparte intreaga ecuatie cu $9^x$ si avem:

$3\left(\dfrac{4}{9}\right)^x-\left(\dfrac{6}{9}\right)^x=2\\ \\ \\ 3\left(\dfrac{2}{3}\right)^{2x}-\left(\dfrac{2}{3}\right)^x=2$

Facem notatia: $t=\left(\dfrac{2}{3}\right)^x,t>0$

Ecuatia devine o ecuatie de grad 2:

$3t^2-t=2\\ \\ 3t^2-t-2=0\\ \Delta = 25\\ \\ t_1=\dfrac{13}{3}, \ \ \ t_2=-4<0$

A doua solutie pica, pentru ca e negativa.

Asadar, avem:

$\left(\dfrac{2}{3}\right)^x=\dfrac{13}{3}\\ \\ \\ x=\log_{\frac{2}{3}}\left(\dfrac{13}{2}\right).$