Вычислите значение выраженияlog5 (10/11) + log25 (242) + log0,2 (40 под корнем)

Используем следующие свойства:

log _(a^m) (c) n = n/m * log_ac.

log_ac + log_ab - log_ad = log_a (c * b : d).

log_aa^m = m.

log₅ (10/11) + log₂₅242 + log_0,2√40 = log₅ (10/11) + log _(5^2) (242) + log_2/10√40 = log₅ (10/11) + log _(5^2) (242) + log_1/5√40 = log₅ (10/11) + log _(5^2) (242) + log _{(5^ (-1))} (√40) = log₅ (10/11) + 1/2 * log₅242 + (1 / (-1)) * log₅√40 = log₅ (10/11) + 1/2 * log₅242 - log₅√40 = log₅ (10/11) + log₅ (242) ^1/2 - log₅√40 = log₅ (10/11 * √242 : √40) = log₅ ((10/11 * 11 * √2) : (2 * √10)) = log₅ (5 * √2/√10) = log₅ (5 * √ (2/10)) = log₅ (5 * √ (1/5)) = log₅ (5 * 5^-1/2) = log₅ (5^1/2) = 1/2.