Expo-equation

**dhiab** · Sep 9th 2009, 11:28 PM

Solve in R : $\frac{{8^x + 27^x }}{{12^x + 18^x }} = \frac{7}{6}<br />$

**red_dog** · Sep 10th 2009, 01:06 AM

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

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

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

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

Divide by $3^{2x}$ :

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

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

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

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

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

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