А знаете ли вы, что... Гриб-шампиньон вырастает всего за 72 часа. (видео)

Логарифмы 2 - возвращение бревен


Прошло уже около трех месяцев с тех пор, как я опубликовал в своем блоге последнюю статью. В ней я дал определение логарифма и рассказал про первое свойство логарифма. Это вторая часть, в которой я расскажу остальные(по крайней мере, самые основные) свойства. В следующей статье я расскажу про все остальное и сделаю все необходимые замечания, которые не вошли в основную статью(например ОДЗ логарифма). Ну, пожалуй, можно приступать.

Разность двух логарифмов Данное свойство аналогично сумме логарифмов, сложение просто заменяется на вычитание, а умножение на деление.

Ну, думаю, что тут все понятно.

Допустим, что у нас есть логарифм некоторого числа b1 по основанию а( или, как я буду в дальнейшем записывать, log (a; b1)) и мы никак не поймем, как нам его вычислить. И тут мы замечаем, что b1 это некоторое число b^n, и если бы не было этой глупой степени, то логарифм вычислялся бы невероятно просто!
Ну ладно, допустим, мы поняли, что log(a; b1) можно записать как log(a; b^n), что дальше? А дальше мы пользуемся следующим свойством:

Доказательство: Допустим, что log(a; b^n) равен некоторому произведению x на n.


Тогда, по определению логарифма.


Извлекая корень n-ной степени из каждой части равенства, получаем


Тогда, если мы в log(a; b) b заменим на a^n и получим log(a; a^n), то будет легко увидеть, что log(a; b) = x


Умножив обе части на n, получаем:


Отсюда получаем, что n * log(a; b) = nx. Но log(a; b^n) тоже равно nx. Значит, n * log(a;b) = log(a; b^n).
Q.E.D.

***

Докажем теперь очень похожее свойство:

Пусть log(a^k; b) = x

Тогда по определению логарифма (a^k)^x=b

Ну или a^kx = b

Тогда log(a; b) = log(a; a^kx) = kx

Поделив все на k получим:
(1/k)*log(a;b) = x. Но log(a^k; b) = x. Из этого следует, что log(a^k; b) = (1/k) * log(a;b)
Q.E.D.

Ок, мы научились избавляться от степеней в логарифмах. Но(по моему мнению, конечно) эта задача ничто по сравнению со сложностью запоминания, когда мы умножаем на ненужную степень, а когда делим на нее. Все, что я смог придумать, так это то, что когда ненужная степень находится внизу(т.е. в основании логарифма), то она там и остается, "внизу под чертой деления".

Ну, думаю на сегодня достаточно, осталось последнее свойство, самое нелюбимое мною в плане запоминания и доказательства, но одно из самых полезных в плане решения всевозможных задач. Но это(и многое другое) будет уже в следующий раз.
Удачи и до следующей статьи!

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