Precalculus Examples

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

Step 1

To write $\frac{1}{1 + \log_{c} (a b)}$ as a fraction with a common denominator, multiply by $\frac{1 + \log_{a} (b c)}{1 + \log_{a} (b c)}$ .

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

Step 2

To write $\frac{1}{1 + \log_{a} (b c)}$ as a fraction with a common denominator, multiply by $\frac{1 + \log_{c} (a b)}{1 + \log_{c} (a b)}$ .

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

Step 3

Write each expression with a common denominator of $(1 + \log_{c} (a b)) (1 + \log_{a} (b c))$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Reorder the factors of $(1 + \log_{c} (a b)) (1 + \log_{a} (b c))$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Add $1$ and $1$ .

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

Step 6

To write $\frac{2 + \log_{a} (b c) + \log_{c} (a b)}{(1 + \log_{a} (b c)) (1 + \log_{c} (a b))}$ as a fraction with a common denominator, multiply by $\frac{1 + \log_{b} (c a)}{1 + \log_{b} (c a)}$ .

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

Step 7

To write $\frac{1}{1 + \log_{b} (c a)}$ as a fraction with a common denominator, multiply by $\frac{(1 + \log_{a} (b c)) (1 + \log_{c} (a b))}{(1 + \log_{a} (b c)) (1 + \log_{c} (a b))}$ .

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

Step 8

Write each expression with a common denominator of $(1 + \log_{a} (b c)) (1 + \log_{c} (a b)) (1 + \log_{b} (c a))$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

Reorder the factors of $(1 + \log_{a} (b c)) (1 + \log_{c} (a b)) (1 + \log_{b} (c a))$ .

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

Step 8.4

Reorder the factors of $(1 + \log_{b} (c a)) ((1 + \log_{a} (b c)) (1 + \log_{c} (a b)))$ .

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Simplify the numerator.

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Step 10.1

Expand $(2 + \log_{a} (b c) + \log_{c} (a b)) (1 + \log_{b} (c a))$ by multiplying each term in the first expression by each term in the second expression.

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

Step 10.2

Simplify each term.

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Step 10.2.1

Multiply $2$ by $1$ .

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

Step 10.2.2

Simplify $2 \log_{b} (c a)$ by moving $2$ inside the logarithm.

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

Step 10.2.3

Apply the product rule to $c a$ .

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

Step 10.2.4

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

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

Step 10.2.5

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

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

Step 10.3

Expand $(1 + \log_{a} (b c)) (1 + \log_{c} (a b))$ using the FOIL Method.

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Step 10.3.1

Apply the distributive property.

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

Step 10.3.2

Apply the distributive property.

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

Step 10.3.3

Apply the distributive property.

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

Step 10.4

Simplify each term.

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Step 10.4.1

Multiply $1$ by $1$ .

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

Step 10.4.2

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

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

Step 10.4.3

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

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

Step 10.5

Add $2$ and $1$ .

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

Step 10.6

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Step 10.7

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Step 10.8

Rewrite $\log_{b} (c^{2} a^{2}) + \log_{a} (b c) \log_{b} (c a) + \log_{c} (a b) \log_{b} (c a) + 3 + \log_{c} (a b (a b)) + \log_{a} (b c (b c)) + \log_{a} (b c) \log_{c} (a b)$ in a factored form.

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Step 10.8.1

Multiply $a$ by $a$ by adding the exponents.

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Step 10.8.1.1

Move $a$ .

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

Step 10.8.1.2

Multiply $a$ by $a$ .

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

Step 10.8.2

Multiply $b$ by $b$ by adding the exponents.

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Step 10.8.2.1

Move $b$ .

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

Step 10.8.2.2

Multiply $b$ by $b$ .

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

Step 10.8.3

Multiply $b$ by $b$ by adding the exponents.

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Step 10.8.3.1

Move $b$ .

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

Step 10.8.3.2

Multiply $b$ by $b$ .

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

Step 10.8.4

Multiply $c$ by $c$ by adding the exponents.

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Step 10.8.4.1

Move $c$ .

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

Step 10.8.4.2

Multiply $c$ by $c$ .

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