Precalculus Examples

$\log (a^{2} - 10 a - 11) - \log (a^{2} - 1)$

Step 1

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

$\log (\frac{a^{2} - 10 a - 11}{a^{2} - 1})$

Step 2

Factor $a^{2} - 10 a - 11$ using the AC method.

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Step 2.1

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 $- 11$ and whose sum is $- 10$ .

$- 11, 1$

Step 2.2

Write the factored form using these integers.

$\log (\frac{(a - 11) (a + 1)}{a^{2} - 1})$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $1$ as $1^{2}$ .

$\log (\frac{(a - 11) (a + 1)}{a^{2} - 1^{2}})$

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 $a = a$ and $b = 1$ .

$\log (\frac{(a - 11) (a + 1)}{(a + 1) (a - 1)})$

Step 4

Cancel the common factor of $a + 1$ .

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Step 4.1

Cancel the common factor.

$\log (\frac{(a - 11) (a + 1)}{(a + 1) (a - 1)})$

Step 4.2

Rewrite the expression.

$\log (\frac{a - 11}{a - 1})$