Algebra Examples

$\frac{1}{\log_{2} (12)} + \frac{1}{\log_{8} (12)} + \frac{1}{\log_{9} (12)}$

Step 1

To write $\frac{1}{\log_{2} (12)}$ as a fraction with a common denominator, multiply by $\frac{\log_{8} (12)}{\log_{8} (12)}$ .

$\frac{1}{\log_{2} (12)} \cdot \frac{\log_{8} (12)}{\log_{8} (12)} + \frac{1}{\log_{8} (12)} + \frac{1}{\log_{9} (12)}$

Step 2

To write $\frac{1}{\log_{8} (12)}$ as a fraction with a common denominator, multiply by $\frac{\log_{2} (12)}{\log_{2} (12)}$ .

$\frac{1}{\log_{2} (12)} \cdot \frac{\log_{8} (12)}{\log_{8} (12)} + \frac{1}{\log_{8} (12)} \cdot \frac{\log_{2} (12)}{\log_{2} (12)} + \frac{1}{\log_{9} (12)}$

Step 3

Write each expression with a common denominator of $\log_{2} (12) \log_{8} (12)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{1}{\log_{2} (12)}$ by $\frac{\log_{8} (12)}{\log_{8} (12)}$ .

$\frac{\log_{8} (12)}{\log_{2} (12) \log_{8} (12)} + \frac{1}{\log_{8} (12)} \cdot \frac{\log_{2} (12)}{\log_{2} (12)} + \frac{1}{\log_{9} (12)}$

Step 3.2

Multiply $\frac{1}{\log_{8} (12)}$ by $\frac{\log_{2} (12)}{\log_{2} (12)}$ .

$\frac{\log_{8} (12)}{\log_{2} (12) \log_{8} (12)} + \frac{\log_{2} (12)}{\log_{8} (12) \log_{2} (12)} + \frac{1}{\log_{9} (12)}$

Step 3.3

Reorder the factors of $\log_{8} (12) \log_{2} (12)$ .

$\frac{\log_{8} (12)}{\log_{2} (12) \log_{8} (12)} + \frac{\log_{2} (12)}{\log_{2} (12) \log_{8} (12)} + \frac{1}{\log_{9} (12)}$

Step 4

Combine the numerators over the common denominator.

$\frac{\log_{8} (12) + \log_{2} (12)}{\log_{2} (12) \log_{8} (12)} + \frac{1}{\log_{9} (12)}$

Step 5

To write $\frac{\log_{8} (12) + \log_{2} (12)}{\log_{2} (12) \log_{8} (12)}$ as a fraction with a common denominator, multiply by $\frac{\log_{9} (12)}{\log_{9} (12)}$ .

$\frac{\log_{8} (12) + \log_{2} (12)}{\log_{2} (12) \log_{8} (12)} \cdot \frac{\log_{9} (12)}{\log_{9} (12)} + \frac{1}{\log_{9} (12)}$

Step 6

To write $\frac{1}{\log_{9} (12)}$ as a fraction with a common denominator, multiply by $\frac{\log_{2} (12) \log_{8} (12)}{\log_{2} (12) \log_{8} (12)}$ .

$\frac{\log_{8} (12) + \log_{2} (12)}{\log_{2} (12) \log_{8} (12)} \cdot \frac{\log_{9} (12)}{\log_{9} (12)} + \frac{1}{\log_{9} (12)} \cdot \frac{\log_{2} (12) \log_{8} (12)}{\log_{2} (12) \log_{8} (12)}$

Step 7

Write each expression with a common denominator of $\log_{2} (12) \log_{8} (12) \log_{9} (12)$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.1

Multiply $\frac{\log_{8} (12) + \log_{2} (12)}{\log_{2} (12) \log_{8} (12)}$ by $\frac{\log_{9} (12)}{\log_{9} (12)}$ .

$\frac{(\log_{8} (12) + \log_{2} (12)) \log_{9} (12)}{\log_{2} (12) \log_{8} (12) \log_{9} (12)} + \frac{1}{\log_{9} (12)} \cdot \frac{\log_{2} (12) \log_{8} (12)}{\log_{2} (12) \log_{8} (12)}$

Step 7.2

Multiply $\frac{1}{\log_{9} (12)}$ by $\frac{\log_{2} (12) \log_{8} (12)}{\log_{2} (12) \log_{8} (12)}$ .

$\frac{(\log_{8} (12) + \log_{2} (12)) \log_{9} (12)}{\log_{2} (12) \log_{8} (12) \log_{9} (12)} + \frac{\log_{2} (12) \log_{8} (12)}{\log_{9} (12) (\log_{2} (12) \log_{8} (12))}$

Step 7.3

Reorder the factors of $\log_{9} (12) (\log_{2} (12) \log_{8} (12))$ .

$\frac{(\log_{8} (12) + \log_{2} (12)) \log_{9} (12)}{\log_{2} (12) \log_{8} (12) \log_{9} (12)} + \frac{\log_{2} (12) \log_{8} (12)}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 8

Combine the numerators over the common denominator.

$\frac{(\log_{8} (12) + \log_{2} (12)) \log_{9} (12) + \log_{2} (12) \log_{8} (12)}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 9

Apply the distributive property.

$\frac{\log_{8} (12) \log_{9} (12) + \log_{2} (12) \log_{9} (12) + \log_{2} (12) \log_{8} (12)}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10

Simplify the numerator.

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Step 10.1

Rewrite $\log_{8} (12)$ using the change of base formula.

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Step 10.1.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\frac{\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{9} (12) + \log_{2} (12) \log_{9} (12) + \log_{2} (12) \log_{8} (12)}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.1.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\frac{\log (12)}{\log (8)} \log_{9} (12) + \log_{2} (12) \log_{9} (12) + \log_{2} (12) \log_{8} (12)}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.2

Rewrite $\log_{9} (12)$ using the change of base formula.

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Step 10.2.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\frac{\frac{\log (12)}{\log (8)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} + \log_{2} (12) \log_{9} (12) + \log_{2} (12) \log_{8} (12)}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.2.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\frac{\log (12)}{\log (8)} \cdot \frac{\log (12)}{\log (9)} + \log_{2} (12) \log_{9} (12) + \log_{2} (12) \log_{8} (12)}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.3

Multiply $\frac{\log (12)}{\log (8)} \cdot \frac{\log (12)}{\log (9)}$ .

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Step 10.3.1

Multiply $\frac{\log (12)}{\log (8)}$ by $\frac{\log (12)}{\log (9)}$ .

$\frac{\frac{\log (12) \log (12)}{\log (8) \log (9)} + \log_{2} (12) \log_{9} (12) + \log_{2} (12) \log_{8} (12)}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.3.2

Raise $\log (12)$ to the power of $1$ .

$\frac{\frac{\log^{1} (12) \log (12)}{\log (8) \log (9)} + \log_{2} (12) \log_{9} (12) + \log_{2} (12) \log_{8} (12)}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.3.3

Raise $\log (12)$ to the power of $1$ .

$\frac{\frac{\log^{1} (12) \log^{1} (12)}{\log (8) \log (9)} + \log_{2} (12) \log_{9} (12) + \log_{2} (12) \log_{8} (12)}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{{\log (12)}^{1 + 1}}{\log (8) \log (9)} + \log_{2} (12) \log_{9} (12) + \log_{2} (12) \log_{8} (12)}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.3.5

Add $1$ and $1$ .

$\frac{\frac{\log^{2} (12)}{\log (8) \log (9)} + \log_{2} (12) \log_{9} (12) + \log_{2} (12) \log_{8} (12)}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.4

Rewrite $\log_{2} (12)$ using the change of base formula.

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Step 10.4.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\frac{\frac{\log^{2} (12)}{\log (8) \log (9)} + \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{9} (12) + \log_{2} (12) \log_{8} (12)}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.4.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\frac{\log^{2} (12)}{\log (8) \log (9)} + \frac{\log (12)}{\log (2)} \log_{9} (12) + \log_{2} (12) \log_{8} (12)}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.5

Rewrite $\log_{9} (12)$ using the change of base formula.

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Step 10.5.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\frac{\frac{\log^{2} (12)}{\log (8) \log (9)} + \frac{\log (12)}{\log (2)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} + \log_{2} (12) \log_{8} (12)}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.5.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\frac{\log^{2} (12)}{\log (8) \log (9)} + \frac{\log (12)}{\log (2)} \cdot \frac{\log (12)}{\log (9)} + \log_{2} (12) \log_{8} (12)}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.6

Multiply $\frac{\log (12)}{\log (2)} \cdot \frac{\log (12)}{\log (9)}$ .

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Step 10.6.1

Multiply $\frac{\log (12)}{\log (2)}$ by $\frac{\log (12)}{\log (9)}$ .

$\frac{\frac{\log^{2} (12)}{\log (8) \log (9)} + \frac{\log (12) \log (12)}{\log (2) \log (9)} + \log_{2} (12) \log_{8} (12)}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.6.2

Raise $\log (12)$ to the power of $1$ .

$\frac{\frac{\log^{2} (12)}{\log (8) \log (9)} + \frac{\log^{1} (12) \log (12)}{\log (2) \log (9)} + \log_{2} (12) \log_{8} (12)}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.6.3

Raise $\log (12)$ to the power of $1$ .

$\frac{\frac{\log^{2} (12)}{\log (8) \log (9)} + \frac{\log^{1} (12) \log^{1} (12)}{\log (2) \log (9)} + \log_{2} (12) \log_{8} (12)}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{\log^{2} (12)}{\log (8) \log (9)} + \frac{{\log (12)}^{1 + 1}}{\log (2) \log (9)} + \log_{2} (12) \log_{8} (12)}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.6.5

Add $1$ and $1$ .

$\frac{\frac{\log^{2} (12)}{\log (8) \log (9)} + \frac{\log^{2} (12)}{\log (2) \log (9)} + \log_{2} (12) \log_{8} (12)}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.7

Rewrite $\log_{2} (12)$ using the change of base formula.

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Step 10.7.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\frac{\frac{\log^{2} (12)}{\log (8) \log (9)} + \frac{\log^{2} (12)}{\log (2) \log (9)} + \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{8} (12)}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.7.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\frac{\log^{2} (12)}{\log (8) \log (9)} + \frac{\log^{2} (12)}{\log (2) \log (9)} + \frac{\log (12)}{\log (2)} \log_{8} (12)}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.8

Rewrite $\log_{8} (12)$ using the change of base formula.

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Step 10.8.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\frac{\frac{\log^{2} (12)}{\log (8) \log (9)} + \frac{\log^{2} (12)}{\log (2) \log (9)} + \frac{\log (12)}{\log (2)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.8.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\frac{\log^{2} (12)}{\log (8) \log (9)} + \frac{\log^{2} (12)}{\log (2) \log (9)} + \frac{\log (12)}{\log (2)} \cdot \frac{\log (12)}{\log (8)}}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.9

Multiply $\frac{\log (12)}{\log (2)} \cdot \frac{\log (12)}{\log (8)}$ .

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Step 10.9.1

Multiply $\frac{\log (12)}{\log (2)}$ by $\frac{\log (12)}{\log (8)}$ .

$\frac{\frac{\log^{2} (12)}{\log (8) \log (9)} + \frac{\log^{2} (12)}{\log (2) \log (9)} + \frac{\log (12) \log (12)}{\log (2) \log (8)}}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.9.2

Raise $\log (12)$ to the power of $1$ .

$\frac{\frac{\log^{2} (12)}{\log (8) \log (9)} + \frac{\log^{2} (12)}{\log (2) \log (9)} + \frac{\log^{1} (12) \log (12)}{\log (2) \log (8)}}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.9.3

Raise $\log (12)$ to the power of $1$ .

$\frac{\frac{\log^{2} (12)}{\log (8) \log (9)} + \frac{\log^{2} (12)}{\log (2) \log (9)} + \frac{\log^{1} (12) \log^{1} (12)}{\log (2) \log (8)}}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.9.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{\log^{2} (12)}{\log (8) \log (9)} + \frac{\log^{2} (12)}{\log (2) \log (9)} + \frac{{\log (12)}^{1 + 1}}{\log (2) \log (8)}}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.9.5

Add $1$ and $1$ .

$\frac{\frac{\log^{2} (12)}{\log (8) \log (9)} + \frac{\log^{2} (12)}{\log (2) \log (9)} + \frac{\log^{2} (12)}{\log (2) \log (8)}}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.10

To write $\frac{\log^{2} (12)}{\log (8) \log (9)}$ as a fraction with a common denominator, multiply by $\frac{\log (2)}{\log (2)}$ .

$\frac{\frac{\log^{2} (12)}{\log (8) \log (9)} \cdot \frac{\log (2)}{\log (2)} + \frac{\log^{2} (12)}{\log (2) \log (9)} + \frac{\log^{2} (12)}{\log (2) \log (8)}}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.11

To write $\frac{\log^{2} (12)}{\log (2) \log (9)}$ as a fraction with a common denominator, multiply by $\frac{\log (8)}{\log (8)}$ .

$\frac{\frac{\log^{2} (12)}{\log (8) \log (9)} \cdot \frac{\log (2)}{\log (2)} + \frac{\log^{2} (12)}{\log (2) \log (9)} \cdot \frac{\log (8)}{\log (8)} + \frac{\log^{2} (12)}{\log (2) \log (8)}}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.12

Write each expression with a common denominator of $\log (8) \log (9) \log (2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.12.1

Multiply $\frac{\log^{2} (12)}{\log (8) \log (9)}$ by $\frac{\log (2)}{\log (2)}$ .

$\frac{\frac{\log^{2} (12) \log (2)}{\log (8) \log (9) \log (2)} + \frac{\log^{2} (12)}{\log (2) \log (9)} \cdot \frac{\log (8)}{\log (8)} + \frac{\log^{2} (12)}{\log (2) \log (8)}}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.12.2

Multiply $\frac{\log^{2} (12)}{\log (2) \log (9)}$ by $\frac{\log (8)}{\log (8)}$ .

$\frac{\frac{\log^{2} (12) \log (2)}{\log (8) \log (9) \log (2)} + \frac{\log^{2} (12) \log (8)}{\log (2) \log (9) \log (8)} + \frac{\log^{2} (12)}{\log (2) \log (8)}}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.12.3

Reorder the factors of $\log (8) \log (9) \log (2)$ .

$\frac{\frac{\log^{2} (12) \log (2)}{\log (2) \log (8) \log (9)} + \frac{\log^{2} (12) \log (8)}{\log (2) \log (9) \log (8)} + \frac{\log^{2} (12)}{\log (2) \log (8)}}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.12.4

Reorder the factors of $\log (2) \log (9) \log (8)$ .

$\frac{\frac{\log^{2} (12) \log (2)}{\log (2) \log (8) \log (9)} + \frac{\log^{2} (12) \log (8)}{\log (2) \log (8) \log (9)} + \frac{\log^{2} (12)}{\log (2) \log (8)}}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.13

Combine the numerators over the common denominator.

$\frac{\frac{\log^{2} (12) \log (2) + \log^{2} (12) \log (8)}{\log (2) \log (8) \log (9)} + \frac{\log^{2} (12)}{\log (2) \log (8)}}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.14

To write $\frac{\log^{2} (12)}{\log (2) \log (8)}$ as a fraction with a common denominator, multiply by $\frac{\log (9)}{\log (9)}$ .

$\frac{\frac{\log^{2} (12) \log (2) + \log^{2} (12) \log (8)}{\log (2) \log (8) \log (9)} + \frac{\log^{2} (12)}{\log (2) \log (8)} \cdot \frac{\log (9)}{\log (9)}}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.15

Multiply $\frac{\log^{2} (12)}{\log (2) \log (8)}$ by $\frac{\log (9)}{\log (9)}$ .

$\frac{\frac{\log^{2} (12) \log (2) + \log^{2} (12) \log (8)}{\log (2) \log (8) \log (9)} + \frac{\log^{2} (12) \log (9)}{\log (2) \log (8) \log (9)}}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 10.16

Combine the numerators over the common denominator.

$\frac{\frac{\log^{2} (12) \log (2) + \log^{2} (12) \log (8) + \log^{2} (12) \log (9)}{\log (2) \log (8) \log (9)}}{\log_{2} (12) \log_{8} (12) \log_{9} (12)}$

Step 11

Simplify the denominator.

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Step 11.1

Rewrite $\log_{2} (12)$ using the change of base formula.

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Step 11.1.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\frac{\frac{\log^{2} (12) \log (2) + \log^{2} (12) \log (8) + \log^{2} (12) \log (9)}{\log (2) \log (8) \log (9)}}{\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{8} (12) \log_{9} (12)}$

Step 11.1.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\frac{\log^{2} (12) \log (2) + \log^{2} (12) \log (8) + \log^{2} (12) \log (9)}{\log (2) \log (8) \log (9)}}{\frac{\log (12)}{\log (2)} \log_{8} (12) \log_{9} (12)}$

Step 11.2

Rewrite $\log_{8} (12)$ using the change of base formula.

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Step 11.2.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\frac{\frac{\log^{2} (12) \log (2) + \log^{2} (12) \log (8) + \log^{2} (12) \log (9)}{\log (2) \log (8) \log (9)}}{\frac{\log (12)}{\log (2)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{9} (12)}$

Step 11.2.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\frac{\log^{2} (12) \log (2) + \log^{2} (12) \log (8) + \log^{2} (12) \log (9)}{\log (2) \log (8) \log (9)}}{\frac{\log (12)}{\log (2)} \cdot \frac{\log (12)}{\log (8)} \log_{9} (12)}$

Step 11.3

Rewrite $\log_{9} (12)$ using the change of base formula.

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Step 11.3.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\frac{\frac{\log^{2} (12) \log (2) + \log^{2} (12) \log (8) + \log^{2} (12) \log (9)}{\log (2) \log (8) \log (9)}}{\frac{\log (12)}{\log (2)} \cdot \frac{\log (12)}{\log (8)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}}$

Step 11.3.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

Step 11.4

Combine exponents.

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Step 11.4.1

Multiply $\frac{\log (12)}{\log (2)}$ by $\frac{\log (12)}{\log (8)}$ .

$\frac{\frac{\log^{2} (12) \log (2) + \log^{2} (12) \log (8) + \log^{2} (12) \log (9)}{\log (2) \log (8) \log (9)}}{\frac{\log (12) \log (12)}{\log (2) \log (8)} \cdot \frac{\log (12)}{\log (9)}}$

Step 11.4.2

Raise $\log (12)$ to the power of $1$ .

$\frac{\frac{\log^{2} (12) \log (2) + \log^{2} (12) \log (8) + \log^{2} (12) \log (9)}{\log (2) \log (8) \log (9)}}{\frac{\log^{1} (12) \log (12)}{\log (2) \log (8)} \cdot \frac{\log (12)}{\log (9)}}$

Step 11.4.3

Raise $\log (12)$ to the power of $1$ .

$\frac{\frac{\log^{2} (12) \log (2) + \log^{2} (12) \log (8) + \log^{2} (12) \log (9)}{\log (2) \log (8) \log (9)}}{\frac{\log^{1} (12) \log^{1} (12)}{\log (2) \log (8)} \cdot \frac{\log (12)}{\log (9)}}$

Step 11.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{\log^{2} (12) \log (2) + \log^{2} (12) \log (8) + \log^{2} (12) \log (9)}{\log (2) \log (8) \log (9)}}{\frac{{\log (12)}^{1 + 1}}{\log (2) \log (8)} \cdot \frac{\log (12)}{\log (9)}}$

Step 11.4.5

Add $1$ and $1$ .

$\frac{\frac{\log^{2} (12) \log (2) + \log^{2} (12) \log (8) + \log^{2} (12) \log (9)}{\log (2) \log (8) \log (9)}}{\frac{\log^{2} (12)}{\log (2) \log (8)} \cdot \frac{\log (12)}{\log (9)}}$

Step 11.4.6

Multiply $\frac{\log^{2} (12)}{\log (2) \log (8)}$ by $\frac{\log (12)}{\log (9)}$ .

$\frac{\frac{\log^{2} (12) \log (2) + \log^{2} (12) \log (8) + \log^{2} (12) \log (9)}{\log (2) \log (8) \log (9)}}{\frac{\log^{2} (12) \log (12)}{\log (2) \log (8) \log (9)}}$

Step 11.4.7

Multiply $\log^{2} (12)$ by $\log (12)$ by adding the exponents.

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Step 11.4.7.1

Multiply $\log^{2} (12)$ by $\log (12)$ .

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Step 11.4.7.1.1

Raise $\log (12)$ to the power of $1$ .

$\frac{\frac{\log^{2} (12) \log (2) + \log^{2} (12) \log (8) + \log^{2} (12) \log (9)}{\log (2) \log (8) \log (9)}}{\frac{\log^{2} (12) \log^{1} (12)}{\log (2) \log (8) \log (9)}}$

Step 11.4.7.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{\log^{2} (12) \log (2) + \log^{2} (12) \log (8) + \log^{2} (12) \log (9)}{\log (2) \log (8) \log (9)}}{\frac{{\log (12)}^{2 + 1}}{\log (2) \log (8) \log (9)}}$

Step 11.4.7.2

Add $2$ and $1$ .

$\frac{\frac{\log^{2} (12) \log (2) + \log^{2} (12) \log (8) + \log^{2} (12) \log (9)}{\log (2) \log (8) \log (9)}}{\frac{\log^{3} (12)}{\log (2) \log (8) \log (9)}}$

Step 12

Multiply the numerator by the reciprocal of the denominator.

$\frac{\log^{2} (12) \log (2) + \log^{2} (12) \log (8) + \log^{2} (12) \log (9)}{\log (2) \log (8) \log (9)} \cdot \frac{\log (2) \log (8) \log (9)}{\log^{3} (12)}$

Step 13

Cancel the common factor of $\log (2) \log (8) \log (9)$ .

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Step 13.1

Cancel the common factor.

$\frac{\log^{2} (12) \log (2) + \log^{2} (12) \log (8) + \log^{2} (12) \log (9)}{\log (2) \log (8) \log (9)} \cdot \frac{\log (2) \log (8) \log (9)}{\log^{3} (12)}$

Step 13.2

Rewrite the expression.

$(\log^{2} (12) \log (2) + \log^{2} (12) \log (8) + \log^{2} (12) \log (9)) \frac{1}{\log^{3} (12)}$

Step 14

Apply the distributive property.

$\log^{2} (12) \log (2) \frac{1}{\log^{3} (12)} + \log^{2} (12) \log (8) \frac{1}{\log^{3} (12)} + \log^{2} (12) \log (9) \frac{1}{\log^{3} (12)}$

Step 15

Simplify.

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Step 15.1

Cancel the common factor of $\log^{2} (12)$ .

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Step 15.1.1

Factor $\log^{2} (12)$ out of $\log^{2} (12) \log (2)$ .

$\log^{2} (12) (\log (2)) \frac{1}{\log^{3} (12)} + \log^{2} (12) \log (8) \frac{1}{\log^{3} (12)} + \log^{2} (12) \log (9) \frac{1}{\log^{3} (12)}$

Step 15.1.2

Factor $\log^{2} (12)$ out of $\log^{3} (12)$ .

$\log^{2} (12) (\log (2)) \frac{1}{\log^{2} (12) \log (12)} + \log^{2} (12) \log (8) \frac{1}{\log^{3} (12)} + \log^{2} (12) \log (9) \frac{1}{\log^{3} (12)}$

Step 15.1.3

Cancel the common factor.

$\log^{2} (12) \log (2) \frac{1}{\log^{2} (12) \log (12)} + \log^{2} (12) \log (8) \frac{1}{\log^{3} (12)} + \log^{2} (12) \log (9) \frac{1}{\log^{3} (12)}$

Step 15.1.4

Rewrite the expression.

$\log (2) \frac{1}{\log (12)} + \log^{2} (12) \log (8) \frac{1}{\log^{3} (12)} + \log^{2} (12) \log (9) \frac{1}{\log^{3} (12)}$

Step 15.2

Combine $\log (2)$ and $\frac{1}{\log (12)}$ .

$\frac{\log (2)}{\log (12)} + \log^{2} (12) \log (8) \frac{1}{\log^{3} (12)} + \log^{2} (12) \log (9) \frac{1}{\log^{3} (12)}$

Step 15.3

Cancel the common factor of $\log^{2} (12)$ .

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Step 15.3.1

Factor $\log^{2} (12)$ out of $\log^{2} (12) \log (8)$ .

$\frac{\log (2)}{\log (12)} + \log^{2} (12) (\log (8)) \frac{1}{\log^{3} (12)} + \log^{2} (12) \log (9) \frac{1}{\log^{3} (12)}$

Step 15.3.2

Factor $\log^{2} (12)$ out of $\log^{3} (12)$ .

$\frac{\log (2)}{\log (12)} + \log^{2} (12) (\log (8)) \frac{1}{\log^{2} (12) \log (12)} + \log^{2} (12) \log (9) \frac{1}{\log^{3} (12)}$

Step 15.3.3

Cancel the common factor.

$\frac{\log (2)}{\log (12)} + \log^{2} (12) \log (8) \frac{1}{\log^{2} (12) \log (12)} + \log^{2} (12) \log (9) \frac{1}{\log^{3} (12)}$

Step 15.3.4

Rewrite the expression.

$\frac{\log (2)}{\log (12)} + \log (8) \frac{1}{\log (12)} + \log^{2} (12) \log (9) \frac{1}{\log^{3} (12)}$

Step 15.4

Combine $\log (8)$ and $\frac{1}{\log (12)}$ .

$\frac{\log (2)}{\log (12)} + \frac{\log (8)}{\log (12)} + \log^{2} (12) \log (9) \frac{1}{\log^{3} (12)}$

Step 15.5

Cancel the common factor of $\log^{2} (12)$ .

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Step 15.5.1

Factor $\log^{2} (12)$ out of $\log^{2} (12) \log (9)$ .

$\frac{\log (2)}{\log (12)} + \frac{\log (8)}{\log (12)} + \log^{2} (12) (\log (9)) \frac{1}{\log^{3} (12)}$

Step 15.5.2

Factor $\log^{2} (12)$ out of $\log^{3} (12)$ .

$\frac{\log (2)}{\log (12)} + \frac{\log (8)}{\log (12)} + \log^{2} (12) (\log (9)) \frac{1}{\log^{2} (12) \log (12)}$

Step 15.5.3

Cancel the common factor.

$\frac{\log (2)}{\log (12)} + \frac{\log (8)}{\log (12)} + \log^{2} (12) \log (9) \frac{1}{\log^{2} (12) \log (12)}$

Step 15.5.4

Rewrite the expression.

$\frac{\log (2)}{\log (12)} + \frac{\log (8)}{\log (12)} + \log (9) \frac{1}{\log (12)}$

Step 15.6

Combine $\log (9)$ and $\frac{1}{\log (12)}$ .

$\frac{\log (2)}{\log (12)} + \frac{\log (8)}{\log (12)} + \frac{\log (9)}{\log (12)}$

Step 16

Combine the numerators over the common denominator.

$\frac{\log (2) + \log (8) + \log (9)}{\log (12)}$

Step 17

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

$\frac{\log (2 \cdot 8) + \log (9)}{\log (12)}$

Step 18

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

$\frac{\log (2 \cdot 8 \cdot 9)}{\log (12)}$

Step 19

Multiply $2 \cdot 8 \cdot 9$ .

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Step 19.1

Multiply $2$ by $8$ .

$\frac{\log (16 \cdot 9)}{\log (12)}$

Step 19.2

Multiply $16$ by $9$ .

$\frac{\log (144)}{\log (12)}$

Step 20

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (144)}{\log (12)}$

Decimal Form:

$2$