1. **State the problem:** Solve the exponential equation $$9^x - 10 \cdot 3^x + 9 = 0$$. 2. **Rewrite the bases:** Note that $$9 = 3^2$$, so we can write $$9^x = (3^2)^x = 3^{2x}$$. 3. **Substitute:** Let $$y = 3^x$$. Then $$9^x = 3^{2x} = (3^x)^2 = y^2$$. 4. **Rewrite the equation:** Substitute into the original equation: $$y^2 - 10y + 9 = 0$$. 5. **Solve the quadratic equation:** Use the quadratic formula or factorization. 6. **Factorization:** $$y^2 - 10y + 9 = (y - 9)(y - 1) = 0$$. 7. **Find roots:** $$y = 9 \quad \text{or} \quad y = 1$$. 8. **Back-substitute:** Recall $$y = 3^x$$, so: - For $$y=9$$: $$3^x = 9 = 3^2 \Rightarrow x = 2$$. - For $$y=1$$: $$3^x = 1 = 3^0 \Rightarrow x = 0$$. **Final answer:** $$x = 0 \text{ or } x = 2$$.