1) 5log b^2/a (a^2/b); если log a (b) = 3 2) log 2 (a^1/3), если log 4 (a^3) = 9 3) lg2.5 если log 4 (125) = a

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Полина Владимирова

Математика

20 сентября, 23:14

1) 5log b^2/a (a^2/b); если log a (b) = 3 2) log 2 (a^1/3), если log 4 (a^3) = 9 3) lg2.5 если log 4 (125) = a

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Ксения Козлова · Answer 1

1. 5 * log_(b²/2) (a²/b) = 5 * (log_(b²/2)a² - log_(b²/2)b) = 5 * (2 * log_(b²/2)a - log_(b²/2)b) = 5 * (2 * (log_aa/log_a (b²/a) - log_ab/log_a (b²/a)) = 5 * (2 * 1 / (log_ab² - log_aa) - log_ab / (log_ab² - log_aa)) = 5 * (2 * 1 / (2 * log_ab - 1) - log_ab / (2 * log_ab - 1) = 5 * (2 * 1 / (2 * 3 - 1) - 3 / (2 * 3 - 1)) = 5 * 2/5 - 5 * 3/5 = 2 - 3 = - 1.

2. log₄a³ = 9 3/2 * log₂a = 9 log₂a = 9 * 2/3 log₂a = 6.

Значит log₂a^1/3 = 1/3 * log₂a = 1/3 * 6 = 2.

3. log₄125 = a 3 * log₄5 = a log₄5 = a/3.

Значит lg2,5 = lg (25/100) = lg25 - lg100 = log₄25/log₄10 - log₄100/log₄10 = (log₄5² - log₄ (5² * 4)) / log₄ (2 * 5) = (2 * log₄5 - (log₄5² + log₄4)) / (log₄2 + log₄5) = (2 * log₄5 - 2 * log₄5 - 1) / (1/2 + a/3) = - 1 / ((3 + 2 * a) / 6) = - 6 / (3 + 2 * a).