Precalculus Examples

$\log_{8} ({(5 \cdot 6^{2} \cdot 11)}^{2})$

Step 1

Expand $\log_{8} ({(5 \cdot 6^{2} \cdot 11)}^{2})$ by moving $2$ outside the logarithm.

$2 \log_{8} (5 \cdot 6^{2} \cdot 11)$

Step 2

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

$2 \log_{8} (5 \cdot 36 \cdot 11)$

Step 3

Multiply $5$ by $36$ .

$2 \log_{8} (180 \cdot 11)$

Step 4

Multiply $180$ by $11$ .

$2 \log_{8} (1980)$

Step 5

Rewrite $\log_{8} (1980)$ as $\log_{8} (2^{2} \cdot 3^{2} \cdot 5 \cdot 11)$ .

$2 \log_{8} (2^{2} \cdot 3^{2} \cdot 5 \cdot 11)$

Step 6

Rewrite $\log_{8} (2^{2} \cdot 3^{2} \cdot 5 \cdot 11)$ as $\log_{8} (2^{2} \cdot 3^{2} \cdot 5) + \log_{8} (11)$ .

$2 (\log_{8} (2^{2} \cdot 3^{2} \cdot 5) + \log_{8} (11))$

Step 7

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

$2 (\log_{8} (4 \cdot 3^{2} \cdot 5) + \log_{8} (11))$

Step 8

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

$2 (\log_{8} (4 \cdot 9 \cdot 5) + \log_{8} (11))$

Step 9

Multiply $4$ by $9$ .

$2 (\log_{8} (36 \cdot 5) + \log_{8} (11))$

Step 10

Multiply $36$ by $5$ .

$2 (\log_{8} (180) + \log_{8} (11))$

Step 11

Simplify each term.

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Step 11.1

Rewrite $\log_{8} (180)$ as $\log_{8} (2^{2} \cdot 3^{2} \cdot 5)$ .

$2 (\log_{8} (2^{2} \cdot 3^{2} \cdot 5) + \log_{8} (11))$

Step 11.2

Rewrite $\log_{8} (2^{2} \cdot 3^{2} \cdot 5)$ as $\log_{8} (2^{2} \cdot 3^{2}) + \log_{8} (5)$ .

$2 (\log_{8} (2^{2} \cdot 3^{2}) + \log_{8} (5) + \log_{8} (11))$

Step 11.3

Simplify each term.

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Step 11.3.1

Rewrite $\log_{8} (2^{2} \cdot 3^{2})$ as $\log_{8} (2^{2}) + \log_{8} (3^{2})$ .

$2 (\log_{8} (2^{2}) + \log_{8} (3^{2}) + \log_{8} (5) + \log_{8} (11))$

Step 11.3.2

Simplify each term.

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Step 11.3.2.1

Expand $\log_{8} (2^{2})$ by moving $2$ outside the logarithm.

$2 (2 \log_{8} (2) + \log_{8} (3^{2}) + \log_{8} (5) + \log_{8} (11))$

Step 11.3.2.2

Logarithm base $8$ of $2$ is $\frac{1}{3}$ .

$2 (2 (\frac{1}{3}) + \log_{8} (3^{2}) + \log_{8} (5) + \log_{8} (11))$

Step 11.3.2.3

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

$2 (\frac{2}{3} + \log_{8} (3^{2}) + \log_{8} (5) + \log_{8} (11))$

Step 11.3.2.4

Expand $\log_{8} (3^{2})$ by moving $2$ outside the logarithm.

$2 (\frac{2}{3} + 2 \log_{8} (3) + \log_{8} (5) + \log_{8} (11))$

Step 12

Apply the distributive property.

$2 (\frac{2}{3}) + 2 (2 \log_{8} (3)) + 2 \log_{8} (5) + 2 \log_{8} (11)$

Step 13

Simplify.

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Step 13.1

Multiply $2 (\frac{2}{3})$ .

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Step 13.1.1

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

$\frac{2 \cdot 2}{3} + 2 (2 \log_{8} (3)) + 2 \log_{8} (5) + 2 \log_{8} (11)$

Step 13.1.2

Multiply $2$ by $2$ .

$\frac{4}{3} + 2 (2 \log_{8} (3)) + 2 \log_{8} (5) + 2 \log_{8} (11)$

Step 13.2

Multiply $2$ by $2$ .

$\frac{4}{3} + 4 \log_{8} (3) + 2 \log_{8} (5) + 2 \log_{8} (11)$