Precalculus Examples

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

Step 1

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

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

Step 2

Simplify the numerator.

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Step 2.1

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

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

Step 2.2

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 $b = 2$ .

$\ln (\frac{(x + 2) (x - 2)}{x + 2})$

Step 3

Cancel the common factor of $x + 2$ .

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Step 3.1

Cancel the common factor.

$\ln (\frac{(x + 2) (x - 2)}{x + 2})$

Step 3.2

Divide $x - 2$ by $1$ .

$\ln (x - 2)$