1. **Problem:** Calculate the value of $$a = 2\log_5 12 - \log_2 8 - 2\log_5 3$$. 2. **Recall logarithm properties:** - $$\log_b (xy) = \log_b x + \log_b y$$ - $$\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$$ - $$\log_b (x^k) = k \log_b x$$ - Change of base formula: $$\log_b a = \frac{\log_c a}{\log_c b}$$ for any positive $$c \neq 1$$. 3. **Evaluate each term:** - $$\log_2 8 = 3$$ because $$2^3 = 8$$. 4. **Rewrite $$2\log_5 12$$ and $$2\log_5 3$$:** - $$2\log_5 12 = \log_5 12^2 = \log_5 144$$ - $$2\log_5 3 = \log_5 3^2 = \log_5 9$$ 5. **Substitute back:** $$a = \log_5 144 - 3 - \log_5 9$$ 6. **Combine the $$\log_5$$ terms:** $$a = \log_5 \left(\frac{144}{9}\right) - 3 = \log_5 16 - 3$$ 7. **Express $$\log_5 16$$ in terms of base 2:** - Since $$16 = 2^4$$, $$\log_5 16 = \log_5 2^4 = 4 \log_5 2$$ 8. **Rewrite $$a$$:** $$a = 4 \log_5 2 - 3$$ 9. **Final answer:** $$\boxed{a = 4 \log_5 2 - 3}$$ This is the simplified exact form of the expression.