Question:

Step 1: Simplify the argument $3-\sqrt{6}$. Observe that: $$3 - \sqrt{6} = (\sqrt{3})^2 - \sqrt{2}\sqrt{3} = \sqrt{3}(\sqrt{3} - \sqrt{2})$$ Step 2: Apply logarithm properties. Using the simplification from Step 1, we have: $$\log_8(3-\sqrt{6}) = \log_8[\sqrt{3}(\sqrt{3}-\sqrt{2})] = \log_8(\sqrt{3}) + \log_8(\sqrt{3}-\sqrt{2})$$ Step 3: Substitute into the original expression. Substitute the result from Step 2 into the given expression: $$4^5 \log_8(3-\sqrt{6}) - 6 \log_8(\sqrt{3}-\sqrt{2})$$ $$= 4^5[\log_8(\sqrt{3}) + \log_8(\sqrt{3}-\sqrt{2})] - 6 \log_8(\sqrt{3}-\sqrt{2})$$ $$= 4^5 \log_8(\sqrt{3}) + 4^5 \log_8(\sqrt{3}-\sqrt{2}) - 6 \log_8(\sqrt{3}-\sqrt{2})$$ $$= 4^5 \log_8(\sqrt{3}) + (4^5 - 6) \log_8(\sqrt{3}-\sqrt{2})$$ Step 4: Calculate $4^5$. $$4^5 = (2^2)^5 = 2^{10} = 1024$$ Step 5: Evaluate the expression. Substitute the value of $4^5$ into the expression from Step 3: $$1024 \log_8(\sqrt{3}) + (1024 - 6) \log_8(\sqrt{3}-\sqrt{2})$$ $$= 1024 \log_8(3^{1/2}) + 1018 \log_8(\sqrt{3}-\sqrt{2})$$ $$= 1024 \cdot \frac{1}{2} \log_8(3) + 1018 \log_8(\sqrt{3}-\sqrt{2})$$ $$= 512 \log_8(3) + 1018 \log_8(\sqrt{3}-\sqrt{2})$$