Precalculus Examples

$\log_{9} ({(12^{2} \cdot 5)}^{5})$

Step 1

Expand $\log_{9} ({(12^{2} \cdot 5)}^{5})$ by moving $5$ outside the logarithm.

$5 \log_{9} (12^{2} \cdot 5)$

Step 2

Raise $12$ to the power of $2$ .

$5 \log_{9} (144 \cdot 5)$

Step 3

Multiply $144$ by $5$ .

$5 \log_{9} (720)$

Step 4

Rewrite $\log_{9} (720)$ as $\log_{9} (2^{4} \cdot 3^{2} \cdot 5)$ .

$5 \log_{9} (2^{4} \cdot 3^{2} \cdot 5)$

Step 5

Rewrite $\log_{9} (2^{4} \cdot 3^{2} \cdot 5)$ as $\log_{9} (2^{4} \cdot 3^{2}) + \log_{9} (5)$ .

$5 (\log_{9} (2^{4} \cdot 3^{2}) + \log_{9} (5))$

Step 6

Raise $2$ to the power of $4$ .

$5 (\log_{9} (16 \cdot 3^{2}) + \log_{9} (5))$

Step 7

Raise $3$ to the power of $2$ .

$5 (\log_{9} (16 \cdot 9) + \log_{9} (5))$

Step 8

Multiply $16$ by $9$ .

$5 (\log_{9} (144) + \log_{9} (5))$

Step 9

Simplify each term.

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Step 9.1

Rewrite $\log_{9} (144)$ as $\log_{9} (2^{4} \cdot 3^{2})$ .

$5 (\log_{9} (2^{4} \cdot 3^{2}) + \log_{9} (5))$

Step 9.2

Rewrite $\log_{9} (2^{4} \cdot 3^{2})$ as $\log_{9} (2^{4}) + \log_{9} (3^{2})$ .

$5 (\log_{9} (2^{4}) + \log_{9} (3^{2}) + \log_{9} (5))$

Step 9.3

Simplify each term.

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Step 9.3.1

Expand $\log_{9} (2^{4})$ by moving $4$ outside the logarithm.

$5 (4 \log_{9} (2) + \log_{9} (3^{2}) + \log_{9} (5))$

Step 9.3.2

Expand $\log_{9} (3^{2})$ by moving $2$ outside the logarithm.

$5 (4 \log_{9} (2) + 2 \log_{9} (3) + \log_{9} (5))$

Step 9.3.3

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

$5 (4 \log_{9} (2) + 2 (\frac{1}{2}) + \log_{9} (5))$

Step 9.3.4

Cancel the common factor of $2$ .

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Step 9.3.4.1

Cancel the common factor.

$5 (4 \log_{9} (2) + 2 (\frac{1}{2}) + \log_{9} (5))$

Step 9.3.4.2

Rewrite the expression.

$5 (4 \log_{9} (2) + 1 + \log_{9} (5))$

Step 10

Apply the distributive property.

$5 (4 \log_{9} (2)) + 5 \cdot 1 + 5 \log_{9} (5)$

Step 11

Simplify.

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Step 11.1

Multiply $4$ by $5$ .

$20 \log_{9} (2) + 5 \cdot 1 + 5 \log_{9} (5)$

Step 11.2

Multiply $5$ by $1$ .

$20 \log_{9} (2) + 5 + 5 \log_{9} (5)$