1. **State the problem:** Simplify and solve for $a$ given the expression: $$a = \ln \left( \left(\sqrt{e+1} - 1\right)^{2025} \right) + \ln \left( \left(\sqrt{e+1} + 1\right)^{2025} \right)$$ 2. **Recall logarithm properties:** - $\ln(x^n) = n \ln(x)$ - $\ln(x) + \ln(y) = \ln(xy)$ 3. **Apply the power rule:** $$a = 2025 \ln(\sqrt{e+1} - 1) + 2025 \ln(\sqrt{e+1} + 1)$$ 4. **Combine the logarithms:** $$a = 2025 \left( \ln(\sqrt{e+1} - 1) + \ln(\sqrt{e+1} + 1) \right) = 2025 \ln \left( (\sqrt{e+1} - 1)(\sqrt{e+1} + 1) \right)$$ 5. **Simplify the product inside the logarithm:** $$(\sqrt{e+1} - 1)(\sqrt{e+1} + 1) = (\sqrt{e+1})^2 - 1^2 = (e+1) - 1 = e$$ 6. **Substitute back:** $$a = 2025 \ln(e)$$ 7. **Recall that $\ln(e) = 1$:** $$a = 2025 \times 1 = 2025$$ **Final answer:** $$\boxed{2025}$$