Основы мат. анализа Примеры

$\frac{1}{1 + \log_{c} (a b)} + \frac{1}{1 + \log_{a} (b c)} + \frac{1}{1 + \log_{b} (c a)}$

Этап 1

Перепишем $\log_{c} (a b)$ в виде $\log_{c} (a) + \log_{c} (b)$ .

$\frac{1}{1 + \log_{c} (a) + \log_{c} (b)} + \frac{1}{1 + \log_{a} (b c)} + \frac{1}{1 + \log_{b} (c a)}$

Этап 2

Перепишем $\log_{a} (b c)$ в виде $\log_{a} (b) + \log_{a} (c)$ .

$\frac{1}{1 + \log_{c} (a) + \log_{c} (b)} + \frac{1}{1 + \log_{a} (b) + \log_{a} (c)} + \frac{1}{1 + \log_{b} (c a)}$

Этап 3

Перепишем $\log_{b} (c a)$ в виде $\log_{b} (c) + \log_{b} (a)$ .

$\frac{1}{1 + \log_{c} (a) + \log_{c} (b)} + \frac{1}{1 + \log_{a} (b) + \log_{a} (c)} + \frac{1}{1 + \log_{b} (c) + \log_{b} (a)}$

Этап 4

Чтобы записать $\frac{1}{1 + \log_{c} (a) + \log_{c} (b)}$ в виде дроби с общим знаменателем, умножим ее на $\frac{1 + \log_{a} (b) + \log_{a} (c)}{1 + \log_{a} (b) + \log_{a} (c)}$ .

$\frac{1}{1 + \log_{c} (a) + \log_{c} (b)} \cdot \frac{1 + \log_{a} (b) + \log_{a} (c)}{1 + \log_{a} (b) + \log_{a} (c)} + \frac{1}{1 + \log_{a} (b) + \log_{a} (c)} + \frac{1}{1 + \log_{b} (c) + \log_{b} (a)}$

Этап 5

Чтобы записать $\frac{1}{1 + \log_{a} (b) + \log_{a} (c)}$ в виде дроби с общим знаменателем, умножим ее на $\frac{1 + \log_{c} (a) + \log_{c} (b)}{1 + \log_{c} (a) + \log_{c} (b)}$ .

$\frac{1}{1 + \log_{c} (a) + \log_{c} (b)} \cdot \frac{1 + \log_{a} (b) + \log_{a} (c)}{1 + \log_{a} (b) + \log_{a} (c)} + \frac{1}{1 + \log_{a} (b) + \log_{a} (c)} \cdot \frac{1 + \log_{c} (a) + \log_{c} (b)}{1 + \log_{c} (a) + \log_{c} (b)} + \frac{1}{1 + \log_{b} (c) + \log_{b} (a)}$

Этап 6

Запишем каждое выражение с общим знаменателем $(1 + \log_{c} (a) + \log_{c} (b)) (1 + \log_{a} (b) + \log_{a} (c))$ , умножив на подходящий множитель $1$ .

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Этап 6.1

Умножим $\frac{1}{1 + \log_{c} (a) + \log_{c} (b)}$ на $\frac{1 + \log_{a} (b) + \log_{a} (c)}{1 + \log_{a} (b) + \log_{a} (c)}$ .

$\frac{1 + \log_{a} (b) + \log_{a} (c)}{(1 + \log_{c} (a) + \log_{c} (b)) (1 + \log_{a} (b) + \log_{a} (c))} + \frac{1}{1 + \log_{a} (b) + \log_{a} (c)} \cdot \frac{1 + \log_{c} (a) + \log_{c} (b)}{1 + \log_{c} (a) + \log_{c} (b)} + \frac{1}{1 + \log_{b} (c) + \log_{b} (a)}$

Этап 6.2

Умножим $\frac{1}{1 + \log_{a} (b) + \log_{a} (c)}$ на $\frac{1 + \log_{c} (a) + \log_{c} (b)}{1 + \log_{c} (a) + \log_{c} (b)}$ .

$\frac{1 + \log_{a} (b) + \log_{a} (c)}{(1 + \log_{c} (a) + \log_{c} (b)) (1 + \log_{a} (b) + \log_{a} (c))} + \frac{1 + \log_{c} (a) + \log_{c} (b)}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))} + \frac{1}{1 + \log_{b} (c) + \log_{b} (a)}$

Этап 6.3

Изменим порядок множителей в $(1 + \log_{c} (a) + \log_{c} (b)) (1 + \log_{a} (b) + \log_{a} (c))$ .

$\frac{1 + \log_{a} (b) + \log_{a} (c)}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))} + \frac{1 + \log_{c} (a) + \log_{c} (b)}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))} + \frac{1}{1 + \log_{b} (c) + \log_{b} (a)}$

Этап 7

Объединим числители над общим знаменателем.

$\frac{1 + \log_{a} (b) + \log_{a} (c) + 1 + \log_{c} (a) + \log_{c} (b)}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))} + \frac{1}{1 + \log_{b} (c) + \log_{b} (a)}$

Этап 8

Добавим $1$ и $1$ .

$\frac{2 + \log_{a} (b) + \log_{a} (c) + \log_{c} (a) + \log_{c} (b)}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))} + \frac{1}{1 + \log_{b} (c) + \log_{b} (a)}$

Этап 9

Чтобы записать $\frac{2 + \log_{a} (b) + \log_{a} (c) + \log_{c} (a) + \log_{c} (b)}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))}$ в виде дроби с общим знаменателем, умножим ее на $\frac{1 + \log_{b} (c) + \log_{b} (a)}{1 + \log_{b} (c) + \log_{b} (a)}$ .

Этап 10

Чтобы записать $\frac{1}{1 + \log_{b} (c) + \log_{b} (a)}$ в виде дроби с общим знаменателем, умножим ее на $\frac{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))}$ .

$\frac{2 + \log_{a} (b) + \log_{a} (c) + \log_{c} (a) + \log_{c} (b)}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))} \cdot \frac{1 + \log_{b} (c) + \log_{b} (a)}{1 + \log_{b} (c) + \log_{b} (a)} + \frac{1}{1 + \log_{b} (c) + \log_{b} (a)} \cdot \frac{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))}$

Этап 11

Запишем каждое выражение с общим знаменателем $(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b)) (1 + \log_{b} (c) + \log_{b} (a))$ , умножив на подходящий множитель $1$ .

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Этап 11.1

Умножим $\frac{2 + \log_{a} (b) + \log_{a} (c) + \log_{c} (a) + \log_{c} (b)}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))}$ на $\frac{1 + \log_{b} (c) + \log_{b} (a)}{1 + \log_{b} (c) + \log_{b} (a)}$ .

$\frac{(2 + \log_{a} (b) + \log_{a} (c) + \log_{c} (a) + \log_{c} (b)) (1 + \log_{b} (c) + \log_{b} (a))}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b)) (1 + \log_{b} (c) + \log_{b} (a))} + \frac{1}{1 + \log_{b} (c) + \log_{b} (a)} \cdot \frac{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))}$

Этап 11.2

Умножим $\frac{1}{1 + \log_{b} (c) + \log_{b} (a)}$ на $\frac{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))}$ .

$\frac{(2 + \log_{a} (b) + \log_{a} (c) + \log_{c} (a) + \log_{c} (b)) (1 + \log_{b} (c) + \log_{b} (a))}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b)) (1 + \log_{b} (c) + \log_{b} (a))} + \frac{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))}{(1 + \log_{b} (c) + \log_{b} (a)) ((1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b)))}$

Этап 11.3

Изменим порядок множителей в $(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b)) (1 + \log_{b} (c) + \log_{b} (a))$ .

$\frac{(2 + \log_{a} (b) + \log_{a} (c) + \log_{c} (a) + \log_{c} (b)) (1 + \log_{b} (c) + \log_{b} (a))}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{b} (c) + \log_{b} (a)) (1 + \log_{c} (a) + \log_{c} (b))} + \frac{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))}{(1 + \log_{b} (c) + \log_{b} (a)) ((1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b)))}$

Этап 11.4

Изменим порядок множителей в $(1 + \log_{b} (c) + \log_{b} (a)) ((1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b)))$ .

$\frac{(2 + \log_{a} (b) + \log_{a} (c) + \log_{c} (a) + \log_{c} (b)) (1 + \log_{b} (c) + \log_{b} (a))}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{b} (c) + \log_{b} (a)) (1 + \log_{c} (a) + \log_{c} (b))} + \frac{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{b} (c) + \log_{b} (a)) (1 + \log_{c} (a) + \log_{c} (b))}$

Этап 12

Объединим числители над общим знаменателем.

$\frac{(2 + \log_{a} (b) + \log_{a} (c) + \log_{c} (a) + \log_{c} (b)) (1 + \log_{b} (c) + \log_{b} (a)) + (1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{b} (c) + \log_{b} (a)) (1 + \log_{c} (a) + \log_{c} (b))}$

Этап 13

Упростим числитель.

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Этап 13.1

Развернем $(2 + \log_{a} (b) + \log_{a} (c) + \log_{c} (a) + \log_{c} (b)) (1 + \log_{b} (c) + \log_{b} (a))$ , умножив каждый член в первом выражении на каждый член во втором выражении.

$\frac{2 \cdot 1 + 2 \log_{b} (c) + 2 \log_{b} (a) + \log_{a} (b) \cdot 1 + \log_{a} (b) \log_{b} (c) + \log_{a} (b) \log_{b} (a) + \log_{a} (c) \cdot 1 + \log_{a} (c) \log_{b} (c) + \log_{a} (c) \log_{b} (a) + \log_{c} (a) \cdot 1 + \log_{c} (a) \log_{b} (c) + \log_{c} (a) \log_{b} (a) + \log_{c} (b) \cdot 1 + \log_{c} (b) \log_{b} (c) + \log_{c} (b) \log_{b} (a) + (1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{b} (c) + \log_{b} (a)) (1 + \log_{c} (a) + \log_{c} (b))}$

Этап 13.2

Упростим каждый член.

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Этап 13.2.1

Умножим $2$ на $1$ .

$\frac{2 + 2 \log_{b} (c) + 2 \log_{b} (a) + \log_{a} (b) \cdot 1 + \log_{a} (b) \log_{b} (c) + \log_{a} (b) \log_{b} (a) + \log_{a} (c) \cdot 1 + \log_{a} (c) \log_{b} (c) + \log_{a} (c) \log_{b} (a) + \log_{c} (a) \cdot 1 + \log_{c} (a) \log_{b} (c) + \log_{c} (a) \log_{b} (a) + \log_{c} (b) \cdot 1 + \log_{c} (b) \log_{b} (c) + \log_{c} (b) \log_{b} (a) + (1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{b} (c) + \log_{b} (a)) (1 + \log_{c} (a) + \log_{c} (b))}$

Этап 13.2.2

Умножим $\log_{a} (b)$ на $1$ .

$\frac{2 + 2 \log_{b} (c) + 2 \log_{b} (a) + \log_{a} (b) + \log_{a} (b) \log_{b} (c) + \log_{a} (b) \log_{b} (a) + \log_{a} (c) \cdot 1 + \log_{a} (c) \log_{b} (c) + \log_{a} (c) \log_{b} (a) + \log_{c} (a) \cdot 1 + \log_{c} (a) \log_{b} (c) + \log_{c} (a) \log_{b} (a) + \log_{c} (b) \cdot 1 + \log_{c} (b) \log_{b} (c) + \log_{c} (b) \log_{b} (a) + (1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{b} (c) + \log_{b} (a)) (1 + \log_{c} (a) + \log_{c} (b))}$

Этап 13.2.3

Умножим $\log_{a} (c)$ на $1$ .

$\frac{2 + 2 \log_{b} (c) + 2 \log_{b} (a) + \log_{a} (b) + \log_{a} (b) \log_{b} (c) + \log_{a} (b) \log_{b} (a) + \log_{a} (c) + \log_{a} (c) \log_{b} (c) + \log_{a} (c) \log_{b} (a) + \log_{c} (a) \cdot 1 + \log_{c} (a) \log_{b} (c) + \log_{c} (a) \log_{b} (a) + \log_{c} (b) \cdot 1 + \log_{c} (b) \log_{b} (c) + \log_{c} (b) \log_{b} (a) + (1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{b} (c) + \log_{b} (a)) (1 + \log_{c} (a) + \log_{c} (b))}$

Этап 13.2.4

Умножим $\log_{c} (a)$ на $1$ .

$\frac{2 + 2 \log_{b} (c) + 2 \log_{b} (a) + \log_{a} (b) + \log_{a} (b) \log_{b} (c) + \log_{a} (b) \log_{b} (a) + \log_{a} (c) + \log_{a} (c) \log_{b} (c) + \log_{a} (c) \log_{b} (a) + \log_{c} (a) + \log_{c} (a) \log_{b} (c) + \log_{c} (a) \log_{b} (a) + \log_{c} (b) \cdot 1 + \log_{c} (b) \log_{b} (c) + \log_{c} (b) \log_{b} (a) + (1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{b} (c) + \log_{b} (a)) (1 + \log_{c} (a) + \log_{c} (b))}$

Этап 13.2.5

Умножим $\log_{c} (b)$ на $1$ .

$\frac{2 + 2 \log_{b} (c) + 2 \log_{b} (a) + \log_{a} (b) + \log_{a} (b) \log_{b} (c) + \log_{a} (b) \log_{b} (a) + \log_{a} (c) + \log_{a} (c) \log_{b} (c) + \log_{a} (c) \log_{b} (a) + \log_{c} (a) + \log_{c} (a) \log_{b} (c) + \log_{c} (a) \log_{b} (a) + \log_{c} (b) + \log_{c} (b) \log_{b} (c) + \log_{c} (b) \log_{b} (a) + (1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{b} (c) + \log_{b} (a)) (1 + \log_{c} (a) + \log_{c} (b))}$

Этап 13.3

Развернем $(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))$ , умножив каждый член в первом выражении на каждый член во втором выражении.

$\frac{2 + 2 \log_{b} (c) + 2 \log_{b} (a) + \log_{a} (b) + \log_{a} (b) \log_{b} (c) + \log_{a} (b) \log_{b} (a) + \log_{a} (c) + \log_{a} (c) \log_{b} (c) + \log_{a} (c) \log_{b} (a) + \log_{c} (a) + \log_{c} (a) \log_{b} (c) + \log_{c} (a) \log_{b} (a) + \log_{c} (b) + \log_{c} (b) \log_{b} (c) + \log_{c} (b) \log_{b} (a) + 1 \cdot 1 + 1 \log_{c} (a) + 1 \log_{c} (b) + \log_{a} (b) \cdot 1 + \log_{a} (b) \log_{c} (a) + \log_{a} (b) \log_{c} (b) + \log_{a} (c) \cdot 1 + \log_{a} (c) \log_{c} (a) + \log_{a} (c) \log_{c} (b)}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{b} (c) + \log_{b} (a)) (1 + \log_{c} (a) + \log_{c} (b))}$

Этап 13.4

Упростим каждый член.

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Этап 13.4.1

Умножим $1$ на $1$ .

$\frac{2 + 2 \log_{b} (c) + 2 \log_{b} (a) + \log_{a} (b) + \log_{a} (b) \log_{b} (c) + \log_{a} (b) \log_{b} (a) + \log_{a} (c) + \log_{a} (c) \log_{b} (c) + \log_{a} (c) \log_{b} (a) + \log_{c} (a) + \log_{c} (a) \log_{b} (c) + \log_{c} (a) \log_{b} (a) + \log_{c} (b) + \log_{c} (b) \log_{b} (c) + \log_{c} (b) \log_{b} (a) + 1 + 1 \log_{c} (a) + 1 \log_{c} (b) + \log_{a} (b) \cdot 1 + \log_{a} (b) \log_{c} (a) + \log_{a} (b) \log_{c} (b) + \log_{a} (c) \cdot 1 + \log_{a} (c) \log_{c} (a) + \log_{a} (c) \log_{c} (b)}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{b} (c) + \log_{b} (a)) (1 + \log_{c} (a) + \log_{c} (b))}$

Этап 13.4.2

Умножим $\log_{c} (a)$ на $1$ .

$\frac{2 + 2 \log_{b} (c) + 2 \log_{b} (a) + \log_{a} (b) + \log_{a} (b) \log_{b} (c) + \log_{a} (b) \log_{b} (a) + \log_{a} (c) + \log_{a} (c) \log_{b} (c) + \log_{a} (c) \log_{b} (a) + \log_{c} (a) + \log_{c} (a) \log_{b} (c) + \log_{c} (a) \log_{b} (a) + \log_{c} (b) + \log_{c} (b) \log_{b} (c) + \log_{c} (b) \log_{b} (a) + 1 + \log_{c} (a) + 1 \log_{c} (b) + \log_{a} (b) \cdot 1 + \log_{a} (b) \log_{c} (a) + \log_{a} (b) \log_{c} (b) + \log_{a} (c) \cdot 1 + \log_{a} (c) \log_{c} (a) + \log_{a} (c) \log_{c} (b)}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{b} (c) + \log_{b} (a)) (1 + \log_{c} (a) + \log_{c} (b))}$

Этап 13.4.3

Умножим $\log_{c} (b)$ на $1$ .

$\frac{2 + 2 \log_{b} (c) + 2 \log_{b} (a) + \log_{a} (b) + \log_{a} (b) \log_{b} (c) + \log_{a} (b) \log_{b} (a) + \log_{a} (c) + \log_{a} (c) \log_{b} (c) + \log_{a} (c) \log_{b} (a) + \log_{c} (a) + \log_{c} (a) \log_{b} (c) + \log_{c} (a) \log_{b} (a) + \log_{c} (b) + \log_{c} (b) \log_{b} (c) + \log_{c} (b) \log_{b} (a) + 1 + \log_{c} (a) + \log_{c} (b) + \log_{a} (b) \cdot 1 + \log_{a} (b) \log_{c} (a) + \log_{a} (b) \log_{c} (b) + \log_{a} (c) \cdot 1 + \log_{a} (c) \log_{c} (a) + \log_{a} (c) \log_{c} (b)}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{b} (c) + \log_{b} (a)) (1 + \log_{c} (a) + \log_{c} (b))}$

Этап 13.4.4

Умножим $\log_{a} (b)$ на $1$ .

$\frac{2 + 2 \log_{b} (c) + 2 \log_{b} (a) + \log_{a} (b) + \log_{a} (b) \log_{b} (c) + \log_{a} (b) \log_{b} (a) + \log_{a} (c) + \log_{a} (c) \log_{b} (c) + \log_{a} (c) \log_{b} (a) + \log_{c} (a) + \log_{c} (a) \log_{b} (c) + \log_{c} (a) \log_{b} (a) + \log_{c} (b) + \log_{c} (b) \log_{b} (c) + \log_{c} (b) \log_{b} (a) + 1 + \log_{c} (a) + \log_{c} (b) + \log_{a} (b) + \log_{a} (b) \log_{c} (a) + \log_{a} (b) \log_{c} (b) + \log_{a} (c) \cdot 1 + \log_{a} (c) \log_{c} (a) + \log_{a} (c) \log_{c} (b)}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{b} (c) + \log_{b} (a)) (1 + \log_{c} (a) + \log_{c} (b))}$

Этап 13.4.5

Умножим $\log_{a} (c)$ на $1$ .

$\frac{2 + 2 \log_{b} (c) + 2 \log_{b} (a) + \log_{a} (b) + \log_{a} (b) \log_{b} (c) + \log_{a} (b) \log_{b} (a) + \log_{a} (c) + \log_{a} (c) \log_{b} (c) + \log_{a} (c) \log_{b} (a) + \log_{c} (a) + \log_{c} (a) \log_{b} (c) + \log_{c} (a) \log_{b} (a) + \log_{c} (b) + \log_{c} (b) \log_{b} (c) + \log_{c} (b) \log_{b} (a) + 1 + \log_{c} (a) + \log_{c} (b) + \log_{a} (b) + \log_{a} (b) \log_{c} (a) + \log_{a} (b) \log_{c} (b) + \log_{a} (c) + \log_{a} (c) \log_{c} (a) + \log_{a} (c) \log_{c} (b)}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{b} (c) + \log_{b} (a)) (1 + \log_{c} (a) + \log_{c} (b))}$

Этап 13.5

Добавим $2$ и $1$ .

$\frac{2 \log_{b} (c) + 2 \log_{b} (a) + \log_{a} (b) + \log_{a} (b) \log_{b} (c) + \log_{a} (b) \log_{b} (a) + \log_{a} (c) + \log_{a} (c) \log_{b} (c) + \log_{a} (c) \log_{b} (a) + \log_{c} (a) + \log_{c} (a) \log_{b} (c) + \log_{c} (a) \log_{b} (a) + \log_{c} (b) + \log_{c} (b) \log_{b} (c) + \log_{c} (b) \log_{b} (a) + 3 + \log_{c} (a) + \log_{c} (b) + \log_{a} (b) + \log_{a} (b) \log_{c} (a) + \log_{a} (b) \log_{c} (b) + \log_{a} (c) + \log_{a} (c) \log_{c} (a) + \log_{a} (c) \log_{c} (b)}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{b} (c) + \log_{b} (a)) (1 + \log_{c} (a) + \log_{c} (b))}$

Этап 13.6

Добавим $\log_{a} (b)$ и $\log_{a} (b)$ .

$\frac{2 \log_{b} (c) + 2 \log_{b} (a) + \log_{a} (b) \log_{b} (c) + \log_{a} (b) \log_{b} (a) + \log_{a} (c) + \log_{a} (c) \log_{b} (c) + \log_{a} (c) \log_{b} (a) + \log_{c} (a) + \log_{c} (a) \log_{b} (c) + \log_{c} (a) \log_{b} (a) + \log_{c} (b) + \log_{c} (b) \log_{b} (c) + \log_{c} (b) \log_{b} (a) + 3 + \log_{c} (a) + \log_{c} (b) + 2 \log_{a} (b) + \log_{a} (b) \log_{c} (a) + \log_{a} (b) \log_{c} (b) + \log_{a} (c) + \log_{a} (c) \log_{c} (a) + \log_{a} (c) \log_{c} (b)}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{b} (c) + \log_{b} (a)) (1 + \log_{c} (a) + \log_{c} (b))}$

Этап 13.7

Добавим $\log_{a} (c)$ и $\log_{a} (c)$ .

$\frac{2 \log_{b} (c) + 2 \log_{b} (a) + \log_{a} (b) \log_{b} (c) + \log_{a} (b) \log_{b} (a) + \log_{a} (c) \log_{b} (c) + \log_{a} (c) \log_{b} (a) + \log_{c} (a) + \log_{c} (a) \log_{b} (c) + \log_{c} (a) \log_{b} (a) + \log_{c} (b) + \log_{c} (b) \log_{b} (c) + \log_{c} (b) \log_{b} (a) + 3 + \log_{c} (a) + \log_{c} (b) + 2 \log_{a} (b) + \log_{a} (b) \log_{c} (a) + \log_{a} (b) \log_{c} (b) + 2 \log_{a} (c) + \log_{a} (c) \log_{c} (a) + \log_{a} (c) \log_{c} (b)}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{b} (c) + \log_{b} (a)) (1 + \log_{c} (a) + \log_{c} (b))}$

Этап 13.8

Добавим $\log_{c} (a)$ и $\log_{c} (a)$ .

$\frac{2 \log_{b} (c) + 2 \log_{b} (a) + \log_{a} (b) \log_{b} (c) + \log_{a} (b) \log_{b} (a) + \log_{a} (c) \log_{b} (c) + \log_{a} (c) \log_{b} (a) + \log_{c} (a) \log_{b} (c) + \log_{c} (a) \log_{b} (a) + \log_{c} (b) + \log_{c} (b) \log_{b} (c) + \log_{c} (b) \log_{b} (a) + 3 + 2 \log_{c} (a) + \log_{c} (b) + 2 \log_{a} (b) + \log_{a} (b) \log_{c} (a) + \log_{a} (b) \log_{c} (b) + 2 \log_{a} (c) + \log_{a} (c) \log_{c} (a) + \log_{a} (c) \log_{c} (b)}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{b} (c) + \log_{b} (a)) (1 + \log_{c} (a) + \log_{c} (b))}$

Этап 13.9

Добавим $\log_{c} (b)$ и $\log_{c} (b)$ .

$\frac{2 \log_{b} (c) + 2 \log_{b} (a) + \log_{a} (b) \log_{b} (c) + \log_{a} (b) \log_{b} (a) + \log_{a} (c) \log_{b} (c) + \log_{a} (c) \log_{b} (a) + \log_{c} (a) \log_{b} (c) + \log_{c} (a) \log_{b} (a) + \log_{c} (b) \log_{b} (c) + \log_{c} (b) \log_{b} (a) + 3 + 2 \log_{c} (a) + 2 \log_{c} (b) + 2 \log_{a} (b) + \log_{a} (b) \log_{c} (a) + \log_{a} (b) \log_{c} (b) + 2 \log_{a} (c) + \log_{a} (c) \log_{c} (a) + \log_{a} (c) \log_{c} (b)}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{b} (c) + \log_{b} (a)) (1 + \log_{c} (a) + \log_{c} (b))}$