Precalculus Examples

$\log_{3} ({(\frac{5}{12^{5}})}^{2})$

Step 1

Expand $\log_{3} ({(\frac{5}{12^{5}})}^{2})$ by moving $2$ outside the logarithm.

$2 \log_{3} (\frac{5}{12^{5}})$

Step 2

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

$2 \log_{3} (\frac{5}{248832})$

Step 3

Rewrite $\log_{3} (\frac{5}{248832})$ as $\log_{3} (5) - \log_{3} (248832)$ .

$2 (\log_{3} (5) - \log_{3} (248832))$

Step 4

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

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

Step 5

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

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

Step 6

Use logarithm rules to move $5$ out of the exponent.

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

Step 7

Logarithm base $3$ of $3$ is $1$ .

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

Step 8

Multiply $5$ by $1$ .

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

Step 9

Expand $\log_{3} (2^{10})$ by moving $10$ outside the logarithm.

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

Step 10

Apply the distributive property.

$2 (\log_{3} (5) - (10 \log_{3} (2)) - 1 \cdot 5)$

Step 11

Multiply $10$ by $- 1$ .

$2 (\log_{3} (5) - 10 \log_{3} (2) - 1 \cdot 5)$

Step 12

Multiply $- 1$ by $5$ .

$2 (\log_{3} (5) - 10 \log_{3} (2) - 5)$

Step 13

Apply the distributive property.

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

Step 14

Simplify.

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Step 14.1

Multiply $- 10$ by $2$ .

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

Step 14.2

Multiply $2$ by $- 5$ .

$2 \log_{3} (5) - 20 \log_{3} (2) - 10$