Precalculus Examples

$\log_{2} (30) - \log_{2} (12) - \log_{2} (40)$

Step 1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{2} (\frac{30}{12}) - \log_{2} (40)$

Step 2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{2} (\frac{\frac{30}{12}}{40})$

Step 3

Cancel the common factor of $30$ and $12$ .

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Step 3.1

Factor $6$ out of $30$ .

$\log_{2} (\frac{\frac{6 (5)}{12}}{40})$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $6$ out of $12$ .

$\log_{2} (\frac{\frac{6 \cdot 5}{6 \cdot 2}}{40})$

Step 3.2.2

Cancel the common factor.

$\log_{2} (\frac{\frac{6 \cdot 5}{6 \cdot 2}}{40})$

Step 3.2.3

Rewrite the expression.

$\log_{2} (\frac{\frac{5}{2}}{40})$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\log_{2} (\frac{5}{2} \cdot \frac{1}{40})$

Step 5

Cancel the common factor of $5$ .

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Step 5.1

Factor $5$ out of $40$ .

$\log_{2} (\frac{5}{2} \cdot \frac{1}{5 (8)})$

Step 5.2

Cancel the common factor.

$\log_{2} (\frac{5}{2} \cdot \frac{1}{5 \cdot 8})$

Step 5.3

Rewrite the expression.

$\log_{2} (\frac{1}{2} \cdot \frac{1}{8})$

Step 6

Multiply $\frac{1}{2}$ by $\frac{1}{8}$ .

$\log_{2} (\frac{1}{2 \cdot 8})$

Step 7

Simplify the expression.

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Step 7.1

Multiply $2$ by $8$ .

$\log_{2} (\frac{1}{16})$

Step 7.2

Logarithm base $2$ of $\frac{1}{16}$ is $- 4$ .

$- 4$