Precalculus Examples

$\log_{4} (x - 4) + 2 \log_{4} (x + 4) - \log_{4} (x^{3} + 2 x^{2} - 16 x - 32)$

Step 1

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

$2 \log_{4} (x + 4) + \log_{4} (\frac{x - 4}{x^{3} + 2 x^{2} - 16 x - 32})$

Step 2

Simplify each term.

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Step 2.1

Simplify $2 \log_{4} (x + 4)$ by moving $2$ inside the logarithm.

$\log_{4} ({(x + 4)}^{2}) + \log_{4} (\frac{x - 4}{x^{3} + 2 x^{2} - 16 x - 32})$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Factor out the greatest common factor from each group.

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Step 2.2.1.1

Group the first two terms and the last two terms.

$\log_{4} ({(x + 4)}^{2}) + \log_{4} (\frac{x - 4}{(x^{3} + 2 x^{2}) - 16 x - 32})$

Step 2.2.1.2

Factor out the greatest common factor (GCF) from each group.

$\log_{4} ({(x + 4)}^{2}) + \log_{4} (\frac{x - 4}{x^{2} (x + 2) - 16 (x + 2)})$

Step 2.2.2

Factor the polynomial by factoring out the greatest common factor, $x + 2$ .

$\log_{4} ({(x + 4)}^{2}) + \log_{4} (\frac{x - 4}{(x + 2) (x^{2} - 16)})$

Step 2.2.3

Rewrite $16$ as $4^{2}$ .

$\log_{4} ({(x + 4)}^{2}) + \log_{4} (\frac{x - 4}{(x + 2) (x^{2} - 4^{2})})$

Step 2.2.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

$\log_{4} ({(x + 4)}^{2}) + \log_{4} (\frac{x - 4}{(x + 2) (x + 4) (x - 4)})$

Step 2.3

Cancel the common factor of $x - 4$ .

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Step 2.3.1

Cancel the common factor.

$\log_{4} ({(x + 4)}^{2}) + \log_{4} (\frac{x - 4}{(x + 2) (x + 4) (x - 4)})$

Step 2.3.2

Rewrite the expression.

$\log_{4} ({(x + 4)}^{2}) + \log_{4} (\frac{1}{(x + 2) (x + 4)})$

Step 3

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

$\log_{4} ({(x + 4)}^{2} \frac{1}{(x + 2) (x + 4)})$

Step 4

Cancel the common factor of $x + 4$ .

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Step 4.1

Factor $x + 4$ out of ${(x + 4)}^{2}$ .

$\log_{4} ((x + 4) (x + 4) \frac{1}{(x + 2) (x + 4)})$

Step 4.2

Factor $x + 4$ out of $(x + 2) (x + 4)$ .

$\log_{4} ((x + 4) (x + 4) \frac{1}{(x + 4) (x + 2)})$

Step 4.3

Cancel the common factor.

$\log_{4} ((x + 4) (x + 4) \frac{1}{(x + 4) (x + 2)})$

Step 4.4

Rewrite the expression.

$\log_{4} ((x + 4) \frac{1}{x + 2})$

Step 5

Multiply $x + 4$ by $\frac{1}{x + 2}$ .

$\log_{4} (\frac{x + 4}{x + 2})$