Precalculus Examples

$\log (\frac{9}{300})$

Step 1

Rewrite $\log (\frac{9}{300})$ as $\log (9) - \log (300)$ .

$\log (9) - \log (300)$

Step 2

Rewrite $\log (300)$ as $\log (2^{2} \cdot 3 \cdot 5^{2})$ .

$\log (9) - \log (2^{2} \cdot 3 \cdot 5^{2})$

Step 3

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

$\log (9) - (\log (2^{2} \cdot 3) + \log (5^{2}))$

Step 4

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

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

Step 5

Multiply $4$ by $3$ .

$\log (9) - (\log (12) + \log (5^{2}))$

Step 6

Simplify each term.

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Step 6.1

Rewrite $\log (12)$ as $\log (2^{2} \cdot 3)$ .

$\log (9) - (\log (2^{2} \cdot 3) + \log (5^{2}))$

Step 6.2

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

$\log (9) - (\log (2^{2}) + \log (3) + \log (5^{2}))$

Step 6.3

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

$\log (9) - (2 \log (2) + \log (3) + \log (5^{2}))$

Step 6.4

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

$\log (9) - (2 \log (2) + \log (3) + 2 \log (5))$

Step 7

Apply the distributive property.

$\log (9) - (2 \log (2)) - \log (3) - (2 \log (5))$

Step 8

Simplify.

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Step 8.1

Multiply $2$ by $- 1$ .

$\log (9) - 2 \log (2) - \log (3) - (2 \log (5))$

Step 8.2

Multiply $2$ by $- 1$ .

$\log (9) - 2 \log (2) - \log (3) - 2 \log (5)$

Step 9

Simplify each term.

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Step 9.1

Rewrite $\log (9)$ as $\log (3^{2})$ .

$\log (3^{2}) - 2 \log (2) - \log (3) - 2 \log (5)$

Step 9.2

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

$2 \log (3) - 2 \log (2) - \log (3) - 2 \log (5)$

Step 10

Subtract $\log (3)$ from $2 \log (3)$ .

$\log (3) - 2 \log (2) - 2 \log (5)$