1. **State the problem:** Simplify the expression $\log_8 27 + \frac{\log_2 5}{\log_4 9} - \log_{\frac{1}{4}} 25$. 2. **Recall the change of base formula:** $$\log_a b = \frac{\log_c b}{\log_c a}$$ for any positive base $c \neq 1$. 3. **Rewrite each term using base 2 logarithms:** - $\log_8 27 = \frac{\log_2 27}{\log_2 8}$ - $\log_4 9 = \frac{\log_2 9}{\log_2 4}$ - $\log_{\frac{1}{4}} 25 = \frac{\log_2 25}{\log_2 \frac{1}{4}}$ 4. **Evaluate the denominators:** - $\log_2 8 = 3$ because $8 = 2^3$ - $\log_2 4 = 2$ because $4 = 2^2$ - $\log_2 \frac{1}{4} = \log_2 4^{-1} = -\log_2 4 = -2$ 5. **Substitute these values:** $$\log_8 27 = \frac{\log_2 27}{3}$$ $$\frac{\log_2 5}{\log_4 9} = \frac{\log_2 5}{\frac{\log_2 9}{2}} = \frac{\log_2 5}{\cancel{\frac{\log_2 9}{2}}} = \log_2 5 \times \frac{2}{\log_2 9}$$ 6. **Simplify the fraction:** $$\frac{\log_2 5}{\log_4 9} = \frac{\log_2 5}{\frac{\log_2 9}{2}} = \log_2 5 \times \frac{2}{\log_2 9}$$ 7. **Rewrite the last term:** $$\log_{\frac{1}{4}} 25 = \frac{\log_2 25}{-2} = -\frac{\log_2 25}{2}$$ 8. **Now the expression is:** $$\frac{\log_2 27}{3} + \log_2 5 \times \frac{2}{\log_2 9} - \frac{\log_2 25}{2}$$ 9. **Express numbers as powers of primes:** - $27 = 3^3 \Rightarrow \log_2 27 = \log_2 3^3 = 3 \log_2 3$ - $9 = 3^2 \Rightarrow \log_2 9 = 2 \log_2 3$ - $25 = 5^2 \Rightarrow \log_2 25 = 2 \log_2 5$ 10. **Substitute these:** $$\frac{3 \log_2 3}{3} + \log_2 5 \times \frac{2}{2 \log_2 3} - \frac{2 \log_2 5}{2}$$ 11. **Simplify:** $$\log_2 3 + \log_2 5 \times \frac{1}{\log_2 3} - \log_2 5$$ 12. **Rewrite the middle term:** $$\log_2 5 \times \frac{1}{\log_2 3} = \frac{\log_2 5}{\log_2 3} = \log_3 5$$ 13. **So the expression is:** $$\log_2 3 + \log_3 5 - \log_2 5$$ 14. **Rewrite $\log_2 5$ as $\frac{\log_2 3 \times \log_3 5}{1}$ to combine terms:** Note that $\log_2 5 = \log_2 3 \times \log_3 5$ because $$\log_2 5 = \frac{\log_3 5}{\log_3 2} = \log_2 3 \times \log_3 5$$ 15. **Substitute:** $$\log_2 3 + \log_3 5 - \log_2 3 \times \log_3 5$$ 16. **Factor:** $$\log_2 3 + \log_3 5 - \log_2 3 \times \log_3 5 = \log_2 3 (1 - \log_3 5) + \log_3 5$$ 17. **Rewrite as:** $$= \log_2 3 - \log_2 3 \times \log_3 5 + \log_3 5$$ 18. **Group:** $$= (\log_2 3 + \log_3 5) - \log_2 3 \times \log_3 5$$ 19. **Recognize this as:** $$(a + b) - ab$$ 20. **No further simplification is straightforward, so this is the simplified form.**