Elementarmathematik Beispiele

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

Schritt 1

Um $\frac{1}{1 + \log_{c} (a b)}$ als Bruch mit einem gemeinsamen Nenner zu schreiben, multipliziere mit $\frac{1 + \log_{a} (b c)}{1 + \log_{a} (b c)}$ .

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

Schritt 2

Um $\frac{1}{1 + \log_{a} (b c)}$ als Bruch mit einem gemeinsamen Nenner zu schreiben, multipliziere mit $\frac{1 + \log_{c} (a b)}{1 + \log_{c} (a b)}$ .

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

Schritt 3

Schreibe jeden Ausdruck mit einem gemeinsamen Nenner von $(1 + \log_{c} (a b)) (1 + \log_{a} (b c))$ , indem du jeden mit einem entsprechenden Faktor von $1$ multiplizierst.

Tippen, um mehr Schritte zu sehen ...

Schritt 3.1

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

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

Schritt 3.2

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

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

Schritt 3.3

Stelle die Faktoren von $(1 + \log_{c} (a b)) (1 + \log_{a} (b c))$ um.

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

Schritt 4

Vereinige die Zähler über dem gemeinsamen Nenner.

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

Schritt 5

Addiere $1$ und $1$ .

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

Schritt 6

Um $\frac{2 + \log_{a} (b c) + \log_{c} (a b)}{(1 + \log_{a} (b c)) (1 + \log_{c} (a b))}$ als Bruch mit einem gemeinsamen Nenner zu schreiben, multipliziere mit $\frac{1 + \log_{b} (c a)}{1 + \log_{b} (c a)}$ .

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

Schritt 7

Um $\frac{1}{1 + \log_{b} (c a)}$ als Bruch mit einem gemeinsamen Nenner zu schreiben, multipliziere mit $\frac{(1 + \log_{a} (b c)) (1 + \log_{c} (a b))}{(1 + \log_{a} (b c)) (1 + \log_{c} (a b))}$ .

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

Schritt 8

Schreibe jeden Ausdruck mit einem gemeinsamen Nenner von $(1 + \log_{a} (b c)) (1 + \log_{c} (a b)) (1 + \log_{b} (c a))$ , indem du jeden mit einem entsprechenden Faktor von $1$ multiplizierst.

Tippen, um mehr Schritte zu sehen ...

Schritt 8.1

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

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

Schritt 8.2

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

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

Schritt 8.3

Stelle die Faktoren von $(1 + \log_{a} (b c)) (1 + \log_{c} (a b)) (1 + \log_{b} (c a))$ um.

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

Schritt 8.4

Stelle die Faktoren von $(1 + \log_{b} (c a)) ((1 + \log_{a} (b c)) (1 + \log_{c} (a b)))$ um.

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

Schritt 9

Vereinige die Zähler über dem gemeinsamen Nenner.

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

Schritt 10

Vereinfache den Zähler.

Tippen, um mehr Schritte zu sehen ...

Schritt 10.1

Multipliziere $(2 + \log_{a} (b c) + \log_{c} (a b)) (1 + \log_{b} (c a))$ aus durch Multiplizieren jedes Terms des ersten Ausdrucks mit jedem Term des zweiten Ausdrucks.

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

Schritt 10.2

Vereinfache jeden Term.

Tippen, um mehr Schritte zu sehen ...

Schritt 10.2.1

Mutltipliziere $2$ mit $1$ .

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

Schritt 10.2.2

Vereinfache $2 \log_{b} (c a)$ , indem du $2$ in den Logarithmus ziehst.

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

Schritt 10.2.3

Wende die Produktregel auf $c a$ an.

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

Schritt 10.2.4

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

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

Schritt 10.2.5

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

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

Schritt 10.3

Multipliziere $(1 + \log_{a} (b c)) (1 + \log_{c} (a b))$ aus unter Verwendung der FOIL-Methode.

Tippen, um mehr Schritte zu sehen ...

Schritt 10.3.1

Wende das Distributivgesetz an.

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

Schritt 10.3.2

Wende das Distributivgesetz an.

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

Schritt 10.3.3

Wende das Distributivgesetz an.

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

Schritt 10.4

Vereinfache jeden Term.

Tippen, um mehr Schritte zu sehen ...

Schritt 10.4.1

Mutltipliziere $1$ mit $1$ .

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

Schritt 10.4.2

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

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

Schritt 10.4.3

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

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

Schritt 10.5

Addiere $2$ und $1$ .

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

Schritt 10.6

Wende die Produktregel für Logarithmen an, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Schritt 10.7

Wende die Produktregel für Logarithmen an, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Schritt 10.8

Schreibe $\log_{b} (c^{2} a^{2}) + \log_{a} (b c) \log_{b} (c a) + \log_{c} (a b) \log_{b} (c a) + 3 + \log_{c} (a b (a b)) + \log_{a} (b c (b c)) + \log_{a} (b c) \log_{c} (a b)$ in eine faktorisierte Form um.

Tippen, um mehr Schritte zu sehen ...

Schritt 10.8.1

Multipliziere $a$ mit $a$ durch Addieren der Exponenten.

Tippen, um mehr Schritte zu sehen ...

Schritt 10.8.1.1

Bewege $a$ .

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

Schritt 10.8.1.2

Mutltipliziere $a$ mit $a$ .

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

Schritt 10.8.2

Multipliziere $b$ mit $b$ durch Addieren der Exponenten.

Tippen, um mehr Schritte zu sehen ...

Schritt 10.8.2.1

Bewege $b$ .

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

Schritt 10.8.2.2

Mutltipliziere $b$ mit $b$ .

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

Schritt 10.8.3

Multipliziere $b$ mit $b$ durch Addieren der Exponenten.

Tippen, um mehr Schritte zu sehen ...

Schritt 10.8.3.1

Bewege $b$ .

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

Schritt 10.8.3.2

Mutltipliziere $b$ mit $b$ .

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

Schritt 10.8.4

Multipliziere $c$ mit $c$ durch Addieren der Exponenten.

Tippen, um mehr Schritte zu sehen ...

Schritt 10.8.4.1

Bewege $c$ .

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

Schritt 10.8.4.2

Mutltipliziere $c$ mit $c$ .

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