1. **State the problem:** Solve for $m$ in the equation $m - n \log_3 2 = 10 \log_9 6$. 2. **Recall logarithm properties:** - Change of base formula: $\log_a b = \frac{\log_c b}{\log_c a}$ for any positive $c \neq 1$. - Relationship between logs with different bases: $\log_9 6$ can be expressed in terms of base 3 since $9 = 3^2$. 3. **Rewrite $\log_9 6$ in terms of base 3:** $$\log_9 6 = \frac{\log_3 6}{\log_3 9} = \frac{\log_3 6}{2}$$ 4. **Substitute back into the equation:** $$m - n \log_3 2 = 10 \times \frac{\log_3 6}{2} = 5 \log_3 6$$ 5. **Express $\log_3 6$ as $\log_3 (2 \times 3)$:** $$\log_3 6 = \log_3 2 + \log_3 3 = \log_3 2 + 1$$ 6. **Substitute this into the equation:** $$m - n \log_3 2 = 5 (\log_3 2 + 1) = 5 \log_3 2 + 5$$ 7. **Isolate $m$:** $$m = n \log_3 2 + 5 \log_3 2 + 5 = (n + 5) \log_3 2 + 5$$ **Final answer:** $$m = (n + 5) \log_3 2 + 5$$