Вычислите log₁₅ (1/3) - log₁₅ (1/5)

Согласно свойству логарифмов log a (b/c) = log a b - log a c.

По правилу деления дробей a/b : c/d = a/b * d/c = (a * d) / (b * c).

Тогда:

log 15 (1/3) - log 15 (1/5) = log 15 ((1/3) / (1/5)) = log 15 ((1/3) * (5/1)) = log 15 ((1 * 5) / (3 * 1)) = log 15 (5/3) = log 15 5 - log 15 3.

Так как по формуле перехода к другому основанию log a b = log c b/log c a, то:

log 15 5 - log 15 3 = log 5 5/log 5 15 - log 3 3/log 3 15 = 1/log 5 (5 * 3) - 1/log 3 (5 * 3).

По свойству логарифмов log a (b * c) = log a b + log a c, тогда:

1/log 5 (5 * 3) - 1/log 3 (5 * 3) = 1 / (log 5 5 + log 5 3) - 1 / (log 3 5 + log 3 3) = 1 / (1 + log 5 3) - 1 / (log 3 5 + 1).

Так как log a b = 1/log b a, то:

1 / (1 + log 5 3) - 1 / (log 3 5 + 1) = 1 / (1 + 1/log 3 5) - 1 / (log 3 5 + 1) = 1 : ((log 3 5 + 1) / log 3 5) - 1 / (log 3 5 + 1) = log 3 5 / (log 3 5 + 1) - 1 / (log 3 5 + 1) = (log 3 5 - 1) / (log 3 5 + 1) = (log 3 5 + 1 - 2) / (log 3 5 + 1) = 1 - 2 / (log 3 5 + 1).