Precalculus Examples

$\log_{4} (x^{2} + 14 x + 45) - \log_{4} (x + 5)$

Step 1

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

$\log_{4} (\frac{x^{2} + 14 x + 45}{x + 5})$

Step 2

Factor $x^{2} + 14 x + 45$ 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 $45$ and whose sum is $14$ .

$5, 9$

Step 2.2

Write the factored form using these integers.

$\log_{4} (\frac{(x + 5) (x + 9)}{x + 5})$

Step 3

Cancel the common factor of $x + 5$ .

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Step 3.1

Cancel the common factor.

$\log_{4} (\frac{(x + 5) (x + 9)}{x + 5})$

Step 3.2

Divide $x + 9$ by $1$ .

$\log_{4} (x + 9)$