Precalculus Examples

$\log (x^{2} - 4) - \log (x^{2} + 9 x + 14)$

Step 1

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

$\log (\frac{x^{2} - 4}{x^{2} + 9 x + 14})$

Step 2

Simplify the numerator.

Tap for more steps...

Step 2.1

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

$\log (\frac{x^{2} - 2^{2}}{x^{2} + 9 x + 14})$

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 $b = 2$ .

$\log (\frac{(x + 2) (x - 2)}{x^{2} + 9 x + 14})$

Step 3

Factor $x^{2} + 9 x + 14$ using the AC method.

Tap for more steps...

Step 3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $14$ and whose sum is $9$ .

$2, 7$

Step 3.2

Write the factored form using these integers.

$\log (\frac{(x + 2) (x - 2)}{(x + 2) (x + 7)})$

Step 4

Cancel the common factor of $x + 2$ .

Tap for more steps...

Step 4.1

Cancel the common factor.

$\log (\frac{(x + 2) (x - 2)}{(x + 2) (x + 7)})$

Step 4.2

Rewrite the expression.

$\log (\frac{x - 2}{x + 7})$