Precalculus Examples

$6 \log_{2} (\frac{2}{3}) + 2 \log_{2} (\frac{1}{6}) - 4 \log_{2} (\frac{2}{9})$

Step 1

Simplify each term.

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Step 1.1

Simplify $6 \log_{2} (\frac{2}{3})$ by moving $6$ inside the logarithm.

$\log_{2} ({(\frac{2}{3})}^{6}) + 2 \log_{2} (\frac{1}{6}) - 4 \log_{2} (\frac{2}{9})$

Step 1.2

Apply the product rule to $\frac{2}{3}$ .

$\log_{2} (\frac{2^{6}}{3^{6}}) + 2 \log_{2} (\frac{1}{6}) - 4 \log_{2} (\frac{2}{9})$

Step 1.3

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

$\log_{2} (\frac{64}{3^{6}}) + 2 \log_{2} (\frac{1}{6}) - 4 \log_{2} (\frac{2}{9})$

Step 1.4

Raise $3$ to the power of $6$ .

$\log_{2} (\frac{64}{729}) + 2 \log_{2} (\frac{1}{6}) - 4 \log_{2} (\frac{2}{9})$

Step 1.5

Simplify $2 \log_{2} (\frac{1}{6})$ by moving $2$ inside the logarithm.

$\log_{2} (\frac{64}{729}) + \log_{2} ({(\frac{1}{6})}^{2}) - 4 \log_{2} (\frac{2}{9})$

Step 1.6

Apply the product rule to $\frac{1}{6}$ .

$\log_{2} (\frac{64}{729}) + \log_{2} (\frac{1^{2}}{6^{2}}) - 4 \log_{2} (\frac{2}{9})$

Step 1.7

One to any power is one.

$\log_{2} (\frac{64}{729}) + \log_{2} (\frac{1}{6^{2}}) - 4 \log_{2} (\frac{2}{9})$

Step 1.8

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

$\log_{2} (\frac{64}{729}) + \log_{2} (\frac{1}{36}) - 4 \log_{2} (\frac{2}{9})$

Step 1.9

Simplify $- 4 \log_{2} (\frac{2}{9})$ by moving $4$ inside the logarithm.

$\log_{2} (\frac{64}{729}) + \log_{2} (\frac{1}{36}) - \log_{2} ({(\frac{2}{9})}^{4})$

Step 1.10

Apply the product rule to $\frac{2}{9}$ .

$\log_{2} (\frac{64}{729}) + \log_{2} (\frac{1}{36}) - \log_{2} (\frac{2^{4}}{9^{4}})$

Step 1.11

Raise $2$ to the power of $4$ .

$\log_{2} (\frac{64}{729}) + \log_{2} (\frac{1}{36}) - \log_{2} (\frac{16}{9^{4}})$

Step 1.12

Raise $9$ to the power of $4$ .

$\log_{2} (\frac{64}{729}) + \log_{2} (\frac{1}{36}) - \log_{2} (\frac{16}{6561})$

Step 2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log_{2} (\frac{64}{729} \cdot \frac{1}{36}) - \log_{2} (\frac{16}{6561})$

Step 3

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

$\log_{2} (\frac{\frac{64}{729} \cdot \frac{1}{36}}{\frac{16}{6561}})$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\log_{2} (\frac{64}{729} \cdot \frac{1}{36} \frac{6561}{16})$

Step 5

Cancel the common factor of $4$ .

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Step 5.1

Factor $4$ out of $64$ .

$\log_{2} (\frac{4 (16)}{729} \cdot \frac{1}{36} \frac{6561}{16})$

Step 5.2

Factor $4$ out of $36$ .

$\log_{2} (\frac{4 \cdot 16}{729} \cdot \frac{1}{4 \cdot 9} \frac{6561}{16})$

Step 5.3

Cancel the common factor.

$\log_{2} (\frac{4 \cdot 16}{729} \cdot \frac{1}{4 \cdot 9} \frac{6561}{16})$

Step 5.4

Rewrite the expression.

$\log_{2} (\frac{16}{729} \cdot \frac{1}{9} \frac{6561}{16})$

Step 6

Multiply $\frac{16}{729}$ by $\frac{1}{9}$ .

$\log_{2} (\frac{16}{729 \cdot 9} \cdot \frac{6561}{16})$

Step 7

Multiply $729$ by $9$ .

$\log_{2} (\frac{16}{6561} \cdot \frac{6561}{16})$

Step 8

Cancel the common factor of $16$ .

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Step 8.1

Cancel the common factor.

$\log_{2} (\frac{16}{6561} \cdot \frac{6561}{16})$

Step 8.2

Rewrite the expression.

$\log_{2} (\frac{1}{6561} \cdot 6561)$

Step 9

Cancel the common factor of $6561$ .

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Step 9.1

Cancel the common factor.

$\log_{2} (\frac{1}{6561} \cdot 6561)$

Step 9.2

Rewrite the expression.

$\log_{2} (1)$

Step 10

Logarithm base $2$ of $1$ is $0$ .

$0$