1. **State the problem:** Simplify the expression $$\log(2)\sqrt{2x+2\sqrt{x^2-1}} + \log(2)(\sqrt{x+1} - \sqrt{x-1})$$. 2. **Recall important formulas and rules:** - The logarithm $$\log(2)$$ is a constant multiplier. - Simplify inside the square roots first. - Use algebraic identities such as $$\sqrt{a} + \sqrt{b}$$ and difference of squares. 3. **Simplify the first square root:** $$\sqrt{2x + 2\sqrt{x^2 - 1}} = \sqrt{2\left(x + \sqrt{x^2 - 1}\right)} = \sqrt{2} \sqrt{x + \sqrt{x^2 - 1}}$$ 4. **Simplify $$\sqrt{x + \sqrt{x^2 - 1}}$$:** Note that $$\sqrt{x + \sqrt{x^2 - 1}} = \sqrt{\frac{(\sqrt{x+1} + \sqrt{x-1})^2}{2}} = \frac{\sqrt{x+1} + \sqrt{x-1}}{\sqrt{2}}$$. 5. **Substitute back:** $$\sqrt{2} \times \frac{\sqrt{x+1} + \sqrt{x-1}}{\sqrt{2}} = \sqrt{x+1} + \sqrt{x-1}$$ 6. **Rewrite the original expression:** $$\log(2)(\sqrt{x+1} + \sqrt{x-1}) + \log(2)(\sqrt{x+1} - \sqrt{x-1})$$ 7. **Factor out $$\log(2)$$:** $$\log(2) \left[(\sqrt{x+1} + \sqrt{x-1}) + (\sqrt{x+1} - \sqrt{x-1})\right] = \log(2) (2\sqrt{x+1})$$ 8. **Final simplified expression:** $$2 \log(2) \sqrt{x+1}$$ **Answer:** $$2 \log(2) \sqrt{x+1}$$