ما قبل التفاضل والتكامل الأمثلة

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

خطوة 1

أعِد كتابة $\log_{c} (a b)$ بالصيغة $\log_{c} (a) + \log_{c} (b)$ .

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

خطوة 2

أعِد كتابة $\log_{a} (b c)$ بالصيغة $\log_{a} (b) + \log_{a} (c)$ .

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

خطوة 3

أعِد كتابة $\log_{b} (c a)$ بالصيغة $\log_{b} (c) + \log_{b} (a)$ .

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

خطوة 4

لكتابة $\frac{1}{1 + \log_{c} (a) + \log_{c} (b)}$ على هيئة كسر بقاسم مشترك، اضرب في $\frac{1 + \log_{a} (b) + \log_{a} (c)}{1 + \log_{a} (b) + \log_{a} (c)}$ .

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

خطوة 5

لكتابة $\frac{1}{1 + \log_{a} (b) + \log_{a} (c)}$ على هيئة كسر بقاسم مشترك، اضرب في $\frac{1 + \log_{c} (a) + \log_{c} (b)}{1 + \log_{c} (a) + \log_{c} (b)}$ .

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

خطوة 6

اكتب كل عبارة قاسمها المشترك $(1 + \log_{c} (a) + \log_{c} (b)) (1 + \log_{a} (b) + \log_{a} (c))$ ، بضربها في العامل المناسب للعدد $1$ .

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خطوة 6.1

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

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

خطوة 6.2

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

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

خطوة 6.3

أعِد ترتيب عوامل $(1 + \log_{c} (a) + \log_{c} (b)) (1 + \log_{a} (b) + \log_{a} (c))$ .

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

خطوة 7

اجمع البسوط على القاسم المشترك.

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

خطوة 8

أضف $1$ و $1$ .

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

خطوة 9

لكتابة $\frac{2 + \log_{a} (b) + \log_{a} (c) + \log_{c} (a) + \log_{c} (b)}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))}$ على هيئة كسر بقاسم مشترك، اضرب في $\frac{1 + \log_{b} (c) + \log_{b} (a)}{1 + \log_{b} (c) + \log_{b} (a)}$ .

خطوة 10

لكتابة $\frac{1}{1 + \log_{b} (c) + \log_{b} (a)}$ على هيئة كسر بقاسم مشترك، اضرب في $\frac{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))}$ .

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

خطوة 11

اكتب كل عبارة قاسمها المشترك $(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b)) (1 + \log_{b} (c) + \log_{b} (a))$ ، بضربها في العامل المناسب للعدد $1$ .

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خطوة 11.1

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

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

خطوة 11.2

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

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

خطوة 11.3

أعِد ترتيب عوامل $(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b)) (1 + \log_{b} (c) + \log_{b} (a))$ .

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

خطوة 11.4

أعِد ترتيب عوامل $(1 + \log_{b} (c) + \log_{b} (a)) ((1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b)))$ .

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

خطوة 12

اجمع البسوط على القاسم المشترك.

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

خطوة 13

بسّط بَسْط الكسر.

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خطوة 13.1

وسّع $(2 + \log_{a} (b) + \log_{a} (c) + \log_{c} (a) + \log_{c} (b)) (1 + \log_{b} (c) + \log_{b} (a))$ بضرب كل حد في العبارة الأولى في كل حد في العبارة الثانية.

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

خطوة 13.2

بسّط كل حد.

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خطوة 13.2.1

اضرب $2$ في $1$ .

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

خطوة 13.2.2

اضرب $\log_{a} (b)$ في $1$ .

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

خطوة 13.2.3

اضرب $\log_{a} (c)$ في $1$ .

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

خطوة 13.2.4

اضرب $\log_{c} (a)$ في $1$ .

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

خطوة 13.2.5

اضرب $\log_{c} (b)$ في $1$ .

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

خطوة 13.3

وسّع $(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))$ بضرب كل حد في العبارة الأولى في كل حد في العبارة الثانية.

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

خطوة 13.4

بسّط كل حد.

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خطوة 13.4.1

اضرب $1$ في $1$ .

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

خطوة 13.4.2

اضرب $\log_{c} (a)$ في $1$ .

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

خطوة 13.4.3

اضرب $\log_{c} (b)$ في $1$ .

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

خطوة 13.4.4

اضرب $\log_{a} (b)$ في $1$ .

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

خطوة 13.4.5

اضرب $\log_{a} (c)$ في $1$ .

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

خطوة 13.5

أضف $2$ و $1$ .

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

خطوة 13.6

أضف $\log_{a} (b)$ و $\log_{a} (b)$ .

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

خطوة 13.7

أضف $\log_{a} (c)$ و $\log_{a} (c)$ .

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

خطوة 13.8

أضف $\log_{c} (a)$ و $\log_{c} (a)$ .

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

خطوة 13.9

أضف $\log_{c} (b)$ و $\log_{c} (b)$ .

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