Precalculus Examples

$\log_{4} (x^{2} - 14 x + 33) - \log_{4} (x - 3)$

Step 1

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

$\log_{4} (\frac{x^{2} - 14 x + 33}{x - 3})$

Step 2

Factor $x^{2} - 14 x + 33$ 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 $33$ and whose sum is $- 14$ .

$- 11, - 3$

Step 2.2

Write the factored form using these integers.

$\log_{4} (\frac{(x - 11) (x - 3)}{x - 3})$

Step 3

Cancel the common factor of $x - 3$ .

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Step 3.1

Cancel the common factor.

$\log_{4} (\frac{(x - 11) (x - 3)}{x - 3})$

Step 3.2

Divide $x - 11$ by $1$ .

$\log_{4} (x - 11)$

Step 4

Rewrite $\log_{4} (x - 11)$ using the change of base formula.

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Step 4.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 4.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (x - 11)}{\log (4)}$