Thread: Expo-equation

Solve in R : $\displaystyle \frac{{8^x + 27^x }}{{12^x + 18^x }} = \frac{7}{6}
$

$\displaystyle \frac{2^{3x}+3^{3x}}{2^{2x}3^x+2^x3^{2x}}=\frac{7} {6}$

$\displaystyle \frac{(2^x+3^x)(2^{2x}-2^x3^x+3^{2x})}{2^x3^x(2^x+3^x)}=\frac{7}{6}$

$\displaystyle \frac{2^{2x}-2^x3^x+3^{2x}}{2^x3^x}=\frac{7}{6}$

$\displaystyle 6\cdot 2^{2x}-132^x3^x+6\cdot 3^{2x}=0$

Divide by $\displaystyle 3^{2x}$:

$\displaystyle 6\left(\frac{2}{3}\right)^{2x}-13\left(\frac{2}{3}\right)^x+6=0$

Let $\displaystyle \left(\frac{2}{3}\right)^x=t$. Then

$\displaystyle 6t^2-13t+6=0\Rightarrow t_1=\frac{2}{3}, \ t_2=\frac{3}{2}$

$\displaystyle \left(\frac{2}{3}\right)^x=\frac{2}{3}\Rightarrow x=1$

$\displaystyle \left(\frac{2}{3}\right)^x=\frac{3}{2}\Rightarrow x=-1$