1. **State the problem:** Solve the equation $$e^{x^2} = e^{7x} \cdot \frac{1}{e^{12}}$$ for $x$. 2. **Rewrite the right side:** Using the property of exponents $$\frac{1}{e^{12}} = e^{-12}$$, so the equation becomes: $$e^{x^2} = e^{7x} \cdot e^{-12} = e^{7x - 12}$$ 3. **Set the exponents equal:** Since the exponential function $e^y$ is one-to-one, we can equate the exponents: $$x^2 = 7x - 12$$ 4. **Rewrite as a quadratic equation:** $$x^2 - 7x + 12 = 0$$ 5. **Factor the quadratic:** $$x^2 - 7x + 12 = (x - 3)(x - 4) = 0$$ 6. **Solve for $x$:** $$x - 3 = 0 \Rightarrow x = 3$$ $$x - 4 = 0 \Rightarrow x = 4$$ 7. **Final answer:** The solution set is $$\boxed{\{3, 4\}}$$.