Question

(log₃4+log₂9)²-(log₃4-log₂9)²
найти значение выражения

Answer 1

$( log_{3}4 + log_{2}9 ) ^{2} - {(log_{3}4 - log_{2}9 )}^{2} = (log_{3}4 + log_{2}9 - log_{3}4 + log_{2}9) \times (log_{3}4 + log_{2}9 + log_{3}4 - log_{2}9 ) = 2 log_{2}(9) \times 2 log_{3}(4) = 2 log_{2}( {3}^{2} ) \times 2 log_{3}( {2}^{2} ) = 4 log_{2}(3) \times 4 log_{3}(2) = 4 \times 4 = 16$

Answer 2

Формула разности квадратов: a² - b² = (a - b)(a + b). 
В нашем случае: (log₃4+log₂9)²-(log₃4-log₂9)² = ((log₃4 + log₂9) - (log₃4-log₂9))((log₃4+log₂9) + (log₃4-log₂9)) = (log₃4 + log₂9 - log₃4 + log₂9)(log₃4 + log₂9 + log₃4 - log₂9) = 2log₂9 * 2log₃4. 
По свойству логарифма: logₐb * logₓc = logₓb * logₐc, xlogₐb = logₐbˣ. 
log₂81 * log₃16 = log₃81 * log₂16 = 4 * 4 = 16. 

Ответ: 16.