Precalculus Examples

$\log_{9} ({(12^{3} \cdot 7)}^{6})$

Step 1

Expand $\log_{9} ({(12^{3} \cdot 7)}^{6})$ by moving $6$ outside the logarithm.

$6 \log_{9} (12^{3} \cdot 7)$

Step 2

Raise $12$ to the power of $3$ .

$6 \log_{9} (1728 \cdot 7)$

Step 3

Multiply $1728$ by $7$ .

$6 \log_{9} (12096)$

Step 4

Rewrite $\log_{9} (12096)$ as $\log_{9} (2^{6} \cdot 3^{3} \cdot 7)$ .

$6 \log_{9} (2^{6} \cdot 3^{3} \cdot 7)$

Step 5

Rewrite $\log_{9} (2^{6} \cdot 3^{3} \cdot 7)$ as $\log_{9} (2^{6} \cdot 3^{3}) + \log_{9} (7)$ .

$6 (\log_{9} (2^{6} \cdot 3^{3}) + \log_{9} (7))$

Step 6

Raise $2$ to the power of $6$ .

$6 (\log_{9} (64 \cdot 3^{3}) + \log_{9} (7))$

Step 7

Raise $3$ to the power of $3$ .

$6 (\log_{9} (64 \cdot 27) + \log_{9} (7))$

Step 8

Multiply $64$ by $27$ .

$6 (\log_{9} (1728) + \log_{9} (7))$

Step 9

Simplify each term.

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Step 9.1

Rewrite $\log_{9} (1728)$ as $\log_{9} (2^{6} \cdot 3^{3})$ .

$6 (\log_{9} (2^{6} \cdot 3^{3}) + \log_{9} (7))$

Step 9.2

Rewrite $\log_{9} (2^{6} \cdot 3^{3})$ as $\log_{9} (2^{6}) + \log_{9} (3^{3})$ .

$6 (\log_{9} (2^{6}) + \log_{9} (3^{3}) + \log_{9} (7))$

Step 9.3

Simplify each term.

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Step 9.3.1

Expand $\log_{9} (2^{6})$ by moving $6$ outside the logarithm.

$6 (6 \log_{9} (2) + \log_{9} (3^{3}) + \log_{9} (7))$

Step 9.3.2

Expand $\log_{9} (3^{3})$ by moving $3$ outside the logarithm.

$6 (6 \log_{9} (2) + 3 \log_{9} (3) + \log_{9} (7))$

Step 9.3.3

Logarithm base $9$ of $3$ is $\frac{1}{2}$ .

$6 (6 \log_{9} (2) + 3 (\frac{1}{2}) + \log_{9} (7))$

Step 9.3.4

Combine $3$ and $\frac{1}{2}$ .

$6 (6 \log_{9} (2) + \frac{3}{2} + \log_{9} (7))$

Step 10

Apply the distributive property.

$6 (6 \log_{9} (2)) + 6 (\frac{3}{2}) + 6 \log_{9} (7)$

Step 11

Simplify.

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Step 11.1

Multiply $6$ by $6$ .

$36 \log_{9} (2) + 6 (\frac{3}{2}) + 6 \log_{9} (7)$

Step 11.2

Cancel the common factor of $2$ .

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Step 11.2.1

Factor $2$ out of $6$ .

$36 \log_{9} (2) + 2 (3) \frac{3}{2} + 6 \log_{9} (7)$

Step 11.2.2

Cancel the common factor.

$36 \log_{9} (2) + 2 \cdot 3 \frac{3}{2} + 6 \log_{9} (7)$

Step 11.2.3

Rewrite the expression.

$36 \log_{9} (2) + 3 \cdot 3 + 6 \log_{9} (7)$

Step 11.3

Multiply $3$ by $3$ .

$36 \log_{9} (2) + 9 + 6 \log_{9} (7)$