Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify the denominator.

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Step 3.1

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

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

Step 3.2

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 $b = 2$ .

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

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $x + 2$ .

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Step 4.1.1

Cancel the common factor.

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

Step 4.1.2

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $x - 2$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

$\ln (1)$

Step 5

The natural logarithm of $1$ is $0$ .

$0$