Precalculus Examples

$\log_{2} (x - 3) + 2 - \log_{2} (x^{2} - 9)$

Step 1

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

$\log_{2} (\frac{x - 3}{x^{2} - 9}) + 2$

Step 2

Simplify each term.

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Step 2.1

Simplify the denominator.

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Step 2.1.1

Rewrite $9$ as $3^{2}$ .

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

Step 2.1.2

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 $b = 3$ .

$\log_{2} (\frac{x - 3}{(x + 3) (x - 3)}) + 2$

Step 2.2

Cancel the common factor of $x - 3$ .

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Step 2.2.1

Cancel the common factor.

$\log_{2} (\frac{x - 3}{(x + 3) (x - 3)}) + 2$

Step 2.2.2

Rewrite the expression.

$\log_{2} (\frac{1}{x + 3}) + 2$