1. **State the problem:** Solve the exponential equation $3^{2x+1} - 5 = 11$ for $x$. 2. **Rewrite the equation:** Add 5 to both sides to isolate the exponential term: $$3^{2x+1} = 16$$ 3. **Apply logarithm:** Take the logarithm base 10 (or natural log) of both sides to bring down the exponent: $$\log(3^{2x+1}) = \log(16)$$ 4. **Use log power rule:** $$ (2x+1) \log(3) = \log(16) $$ 5. **Solve for $x$:** $$ 2x + 1 = \frac{\log(16)}{\log(3)} $$ Calculate the right side: $$ \frac{\log(16)}{\log(3)} \approx 2.52 $$ 6. **Isolate $x$:** $$ 2x = 2.52 - 1 = 1.52 $$ $$ x = \frac{1.52}{2} = 0.76 $$ **Final answer:** $$ x = 0.76 $$