Precalculus Examples

$3 \log (4) - \log (8) + 4 \log (2) - 2 \log (8)$

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \log (4)$ by moving $3$ inside the logarithm.

$\log (4^{3}) - \log (8) + 4 \log (2) - 2 \log (8)$

Step 1.2

Raise $4$ to the power of $3$ .

$\log (64) - \log (8) + 4 \log (2) - 2 \log (8)$

Step 1.3

Simplify $4 \log (2)$ by moving $4$ inside the logarithm.

$\log (64) - \log (8) + \log (2^{4}) - 2 \log (8)$

Step 1.4

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

$\log (64) - \log (8) + \log (16) - 2 \log (8)$

Step 1.5

Simplify $- 2 \log (8)$ by moving $2$ inside the logarithm.

$\log (64) - \log (8) + \log (16) - \log (8^{2})$

Step 1.6

Raise $8$ to the power of $2$ .

$\log (64) - \log (8) + \log (16) - \log (64)$

Step 2

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

$\log (\frac{64}{8}) + \log (16) - \log (64)$

Step 3

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

$\log (\frac{64}{8} \cdot 16) - \log (64)$

Step 4

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

$\log (\frac{\frac{64}{8} \cdot 16}{64})$

Step 5

Cancel the common factor of $64$ and $8$ .

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Step 5.1

Factor $8$ out of $64$ .

$\log (\frac{\frac{8 \cdot 8}{8} \cdot 16}{64})$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $8$ out of $8$ .

$\log (\frac{\frac{8 \cdot 8}{8 (1)} \cdot 16}{64})$

Step 5.2.2

Cancel the common factor.

$\log (\frac{\frac{8 \cdot 8}{8 \cdot 1} \cdot 16}{64})$

Step 5.2.3

Rewrite the expression.

$\log (\frac{\frac{8}{1} \cdot 16}{64})$

Step 5.2.4

Divide $8$ by $1$ .

$\log (\frac{8 \cdot 16}{64})$

Step 6

Cancel the common factor of $8$ and $64$ .

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Step 6.1

Factor $8$ out of $8 \cdot 16$ .

$\log (\frac{8 (16)}{64})$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $8$ out of $64$ .

$\log (\frac{8 \cdot 16}{8 \cdot 8})$

Step 6.2.2

Cancel the common factor.

$\log (\frac{8 \cdot 16}{8 \cdot 8})$

Step 6.2.3

Rewrite the expression.

$\log (\frac{16}{8})$

Step 7

Cancel the common factor of $16$ and $8$ .

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Step 7.1

Factor $8$ out of $16$ .

$\log (\frac{8 \cdot 2}{8})$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $8$ out of $8$ .

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

Step 7.2.2

Cancel the common factor.

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

Step 7.2.3

Rewrite the expression.

$\log (\frac{2}{1})$

Step 7.2.4

Divide $2$ by $1$ .

$\log (2)$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\log (2)$

Decimal Form:

$0.30102999 \dots$