Answer
$\displaystyle \{\log_{3}(\frac{3-\sqrt{5}}{2}), \ \log_{3}(\frac{3+\sqrt{5}}{2})\}$ or $\{-0.876,\ 0.876\}$
Work Step by Step
...$ \quad 9^{x}=(3^{2})^{x}=(3^{x})^{2}$
...$ \quad 3^{x+1}=3^{x}\cdot 3$
... Substitute $t=3^{x}$ (t is positive)
... the equation becomes
$t^{2}-3t+1=0 \quad $...$ \quad $ quadratic formula...
$t=\displaystyle \frac{3\pm\sqrt{(-3)^{2}-4(1)(1)}}{2(1)}=\frac{3\pm\sqrt{5}}{2}$
... both solutions for t are positive, therefore acceptable.
$3^{x}=\displaystyle \frac{3\pm\sqrt{5}}{2} \quad $...Apply$: \quad \log_{3}(...)$
$x=\displaystyle \log_{3}(\frac{3\pm\sqrt{5}}{2})$
$\displaystyle \log_{3}(\frac{3+\sqrt{5}}{2})\approx 0.876$
$\displaystyle \log_{3}(\frac{3-\sqrt{5}}{2})\approx-0.876$
The solution set is $\displaystyle \{\log_{3}(\frac{3-\sqrt{5}}{2}), \ \log_{3}(\frac{3+\sqrt{5}}{2})\}$ or $\{-0.876,\ 0.876\}$
