Precalculus Examples

$\log_{2} ({(12^{5} \cdot 7)}^{2})$

Step 1

Expand $\log_{2} ({(12^{5} \cdot 7)}^{2})$ by moving $2$ outside the logarithm.

$2 \log_{2} (12^{5} \cdot 7)$

Step 2

Raise $12$ to the power of $5$ .

$2 \log_{2} (248832 \cdot 7)$

Step 3

Multiply $248832$ by $7$ .

$2 \log_{2} (1741824)$

Step 4

Rewrite $\log_{2} (1741824)$ as $\log_{2} (2^{10} \cdot 3^{5} \cdot 7)$ .

$2 \log_{2} (2^{10} \cdot 3^{5} \cdot 7)$

Step 5

Rewrite $\log_{2} (2^{10} \cdot 3^{5} \cdot 7)$ as $\log_{2} (2^{10} \cdot 3^{5}) + \log_{2} (7)$ .

$2 (\log_{2} (2^{10} \cdot 3^{5}) + \log_{2} (7))$

Step 6

Raise $2$ to the power of $10$ .

$2 (\log_{2} (1024 \cdot 3^{5}) + \log_{2} (7))$

Step 7

Raise $3$ to the power of $5$ .

$2 (\log_{2} (1024 \cdot 243) + \log_{2} (7))$

Step 8

Multiply $1024$ by $243$ .

$2 (\log_{2} (248832) + \log_{2} (7))$

Step 9

Simplify each term.

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Step 9.1

Rewrite $\log_{2} (248832)$ as $\log_{2} (2^{10} \cdot 3^{5})$ .

$2 (\log_{2} (2^{10} \cdot 3^{5}) + \log_{2} (7))$

Step 9.2

Rewrite $\log_{2} (2^{10} \cdot 3^{5})$ as $\log_{2} (2^{10}) + \log_{2} (3^{5})$ .

$2 (\log_{2} (2^{10}) + \log_{2} (3^{5}) + \log_{2} (7))$

Step 9.3

Use logarithm rules to move $10$ out of the exponent.

$2 (10 \log_{2} (2) + \log_{2} (3^{5}) + \log_{2} (7))$

Step 9.4

Logarithm base $2$ of $2$ is $1$ .

$2 (10 \cdot 1 + \log_{2} (3^{5}) + \log_{2} (7))$

Step 9.5

Multiply $10$ by $1$ .

$2 (10 + \log_{2} (3^{5}) + \log_{2} (7))$

Step 9.6

Expand $\log_{2} (3^{5})$ by moving $5$ outside the logarithm.

$2 (10 + 5 \log_{2} (3) + \log_{2} (7))$

Step 10

Apply the distributive property.

$2 \cdot 10 + 2 (5 \log_{2} (3)) + 2 \log_{2} (7)$

Step 11

Simplify.

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Step 11.1

Multiply $2$ by $10$ .

$20 + 2 (5 \log_{2} (3)) + 2 \log_{2} (7)$

Step 11.2

Multiply $5$ by $2$ .

$20 + 10 \log_{2} (3) + 2 \log_{2} (7)$