81^(log ₉ 2 - 0,25log ₃ 2) =

Сначала упростим показатель степени. Используем свойства логарифмов: n * logₐ b = logₐ (bⁿ) и logₐ b - logₐ c = logₐ (b/c), а также свойство смены основания logₐ b = logₓ b / logₓ a. log₉ 2 - 0.25 log₃ 2 = log₉ 2 - (1/4) log₃ 2 = log₉ 2 - log₃ (2^(1/4)) = log₉ 2 - log₃ √[4]{2} = log₃ 2 / log₃ 9 - log₃ √[4]{2} = log₃ 2 / 2 - log₃ √[4]{2} = (1/2) log₃ 2 - log₃ √[4]{2} = log₃ √2 - log₃ √[4]{2} = log₃ (√2 / √[4]{2}) = log₃ (2^(1/2) / 2^(1/4)) = log₃ (2^(1/2 - 1/4)) = log₃ (2^(1/4)) = log₃ √[4]{2} Теперь вычислим исходное выражение: 81^(log₉ 2 - 0.25 log₃ 2) = 81^(log₃ √[4]{2}) = (3⁴)^(log₃ √[4]{2}) = 3^(4 * log₃ √[4]{2}) = 3^(log₃ (√[4]{2})⁴) = 3^(log₃ (2^(1/4))⁴) = 3^(log₃ 2) = 2