Precalcolo Esempi

\frac{1}{1 + {log}_{c} (a b)} + \frac{1}{1 + {log}_{a} (b c)} + \frac{1}{1 + {log}_{b} (c a)}

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

Passaggio 1

Riscrivi

$\log_{c} (a b)$ come

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

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

Passaggio 2

Riscrivi

$\log_{a} (b c)$ come

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

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

Passaggio 3

Riscrivi

$\log_{b} (c a)$ come

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

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

Passaggio 4

Per scrivere

$\frac{1}{1 + \log_{c} (a) + \log_{c} (b)}$ come una frazione con un comune denominatore, moltiplicala per

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

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

Passaggio 5

Per scrivere

$\frac{1}{1 + \log_{a} (b) + \log_{a} (c)}$ come una frazione con un comune denominatore, moltiplicala per

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

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

Passaggio 6

Scrivi ogni espressione con un comune denominatore di

$(1 + \log_{c} (a) + \log_{c} (b)) (1 + \log_{a} (b) + \log_{a} (c))$ , moltiplicando ciascuna per il fattore appropriato di

$1$ .

Tocca per altri passaggi...

Passaggio 6.1

Moltiplica

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

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

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

Passaggio 6.2

Moltiplica

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

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

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

Passaggio 6.3

Riordina i fattori di

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

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

Passaggio 7

Riduci i numeratori su un comune denominatore.

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

Passaggio 8

Somma

$1$ e

$1$ .

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

Passaggio 9

Per scrivere

$\frac{2 + \log_{a} (b) + \log_{a} (c) + \log_{c} (a) + \log_{c} (b)}{(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))}$ come una frazione con un comune denominatore, moltiplicala per

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

Passaggio 10

Per scrivere

$\frac{1}{1 + \log_{b} (c) + \log_{b} (a)}$ come una frazione con un comune denominatore, moltiplicala per

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

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

Passaggio 11

Scrivi ogni espressione con un comune denominatore di

$(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b)) (1 + \log_{b} (c) + \log_{b} (a))$ , moltiplicando ciascuna per il fattore appropriato di

$1$ .

Tocca per altri passaggi...

Passaggio 11.1

Moltiplica

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

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

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

Passaggio 11.2

Moltiplica

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

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

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

Passaggio 11.3

Riordina i fattori di

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

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

Passaggio 11.4

Riordina i fattori di

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

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

Passaggio 12

Riduci i numeratori su un comune denominatore.

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

Passaggio 13

Semplifica il numeratore.

Tocca per altri passaggi...

Passaggio 13.1

Espandi

$(2 + \log_{a} (b) + \log_{a} (c) + \log_{c} (a) + \log_{c} (b)) (1 + \log_{b} (c) + \log_{b} (a))$ moltiplicando ciascun termine della prima espressione per ciascun termine della seconda espressione.

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

Passaggio 13.2

Semplifica ciascun termine.

Tocca per altri passaggi...

Passaggio 13.2.1

Moltiplica

$2$ per

$1$ .

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

Passaggio 13.2.2

Moltiplica

$\log_{a} (b)$ per

$1$ .

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

Passaggio 13.2.3

Moltiplica

$\log_{a} (c)$ per

$1$ .

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

Passaggio 13.2.4

Moltiplica

$\log_{c} (a)$ per

$1$ .

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

Passaggio 13.2.5

Moltiplica

$\log_{c} (b)$ per

$1$ .

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

Passaggio 13.3

Espandi

$(1 + \log_{a} (b) + \log_{a} (c)) (1 + \log_{c} (a) + \log_{c} (b))$ moltiplicando ciascun termine della prima espressione per ciascun termine della seconda espressione.

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

Passaggio 13.4

Semplifica ciascun termine.

Tocca per altri passaggi...

Passaggio 13.4.1

Moltiplica

$1$ per

$1$ .

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

Passaggio 13.4.2

Moltiplica

$\log_{c} (a)$ per

$1$ .

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

Passaggio 13.4.3

Moltiplica

$\log_{c} (b)$ per

$1$ .

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

Passaggio 13.4.4

Moltiplica

$\log_{a} (b)$ per

$1$ .

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

Passaggio 13.4.5

Moltiplica

$\log_{a} (c)$ per

$1$ .

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

Passaggio 13.5

Somma

$2$ e

$1$ .

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

Passaggio 13.6

Somma

$\log_{a} (b)$ e

$\log_{a} (b)$ .

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

Passaggio 13.7

Somma

$\log_{a} (c)$ e

$\log_{a} (c)$ .

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

Passaggio 13.8

Somma

$\log_{c} (a)$ e

$\log_{c} (a)$ .

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

Passaggio 13.9

Somma

$\log_{c} (b)$ e

$\log_{c} (b)$ .

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