1. The problem asks us to evaluate $\log_6 4^8$ without a calculator. 2. Recall the logarithm power rule: $\log_b (a^n) = n \log_b a$. 3. Applying this rule, we get: $$\log_6 4^8 = 8 \log_6 4$$ 4. Next, express 4 as $2^2$ to simplify the logarithm: $$8 \log_6 4 = 8 \log_6 (2^2)$$ 5. Using the power rule again: $$8 \times 2 \log_6 2 = 16 \log_6 2$$ 6. Since $\log_6 2$ cannot be simplified further without a calculator, the exact expression is: $$16 \log_6 2$$ 7. However, the problem asks for an integer or fraction. Since $\log_6 4^8 = \log_6 (4^8)$, and $4^8 = (2^2)^8 = 2^{16}$, we can rewrite: $$\log_6 4^8 = \log_6 2^{16} = 16 \log_6 2$$ 8. Without further simplification, the answer is $16 \log_6 2$. Final answer: $16 \log_6 2$