Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $x$ .

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Step 4.1.1

Cancel the common factor.

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

Step 4.1.2

Rewrite the expression.

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

Step 4.2

Multiply $x^{2} - 4$ by $1$ .

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

Step 5

Simplify the numerator.

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Step 5.1

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

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

Step 5.2

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 $b = 2$ .

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

Step 6

Cancel the common factor of $x + 2$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Divide $x - 2$ by $1$ .

$\log (x - 2)$