12. 2log₁/₃ 6 - 1/2 log₁/₃ 400 - 3 log₃ √45.

12. Вычислим значение выражения 2log₁/₃ 6 - 1/2 log₁/₃ 400 - 3 log₃ √45.

2log₁/₃ 6 - 1/2 log₁/₃ 400 - 3 log₃ √45 = log₁/₃ 6² - log₁/₃ √400 - 3 log₃ (45)^(1/2) = log₁/₃ 36 - log₁/₃ 20 - log₃ (45)^(3/2) = log₁/₃ (36/20) - log₃ (45)^(3/2) = log₁/₃ (9/5) - log₃ (45)^(3/2) = log₃⁻¹ (9/5) - log₃ (45)^(3/2) = - log₃ (9/5) - log₃ (45)^(3/2) = - log₃ (9/5 * (45)^(3/2)) = - log₃ (9/5 * (9 * 5)^(3/2)) = - log₃ (9/5 * 9^(3/2) * 5^(3/2)) = - log₃ (9^(5/2) * 5^(1/2)) = - log₃ (3^(5) * 5^(1/2)) = - (log₃ 3⁵ + log₃ √5) = - (5 + log₃ √5)

Ответ: -5 - log₃ √5