1. **State the problem:** Solve the exponential equation $$7^{x+1} + 2 \cdot 7^x = 11$$. 2. **Rewrite the equation:** Use the property of exponents $$7^{x+1} = 7^x \cdot 7$$ to factor the expression: $$7^x \cdot 7 + 2 \cdot 7^x = 11$$ 3. **Factor out $$7^x$$:** $$7^x (7 + 2) = 11$$ 4. **Simplify inside the parentheses:** $$7^x \cdot 9 = 11$$ 5. **Isolate $$7^x$$:** $$7^x = \frac{11}{9}$$ 6. **Take the logarithm base 7 of both sides:** $$\log_7 7^x = \log_7 \left(\frac{11}{9}\right)$$ 7. **Use the logarithm power rule:** $$x \cdot \log_7 7 = \log_7 \left(\frac{11}{9}\right)$$ Since $$\log_7 7 = 1$$, this simplifies to: $$x = \log_7 \left(\frac{11}{9}\right)$$ 8. **Express in terms of common logarithms if needed:** $$x = \frac{\log \left(\frac{11}{9}\right)}{\log 7}$$ **Final answer:** $$x = \log_7 \left(\frac{11}{9}\right) = \frac{\log \left(\frac{11}{9}\right)}{\log 7}$$