Precalculus Examples

$\log_{7} (x^{2} - 11 x + 30) - \log_{7} (x^{2} - 36) + \log_{7} (x + 6)$

Step 1

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

$\log_{7} (\frac{x^{2} - 11 x + 30}{x^{2} - 36}) + \log_{7} (x + 6)$

Step 2

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

$\log_{7} (\frac{x^{2} - 11 x + 30}{x^{2} - 36} (x + 6))$

Step 3

Factor $x^{2} - 11 x + 30$ 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 $30$ and whose sum is $- 11$ .

$- 6, - 5$

Step 3.2

Write the factored form using these integers.

$\log_{7} (\frac{(x - 6) (x - 5)}{x^{2} - 36} (x + 6))$

Step 4

Simplify the denominator.

Tap for more steps...

Step 4.1

Rewrite $36$ as $6^{2}$ .

$\log_{7} (\frac{(x - 6) (x - 5)}{x^{2} - 6^{2}} (x + 6))$

Step 4.2

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 $b = 6$ .

$\log_{7} (\frac{(x - 6) (x - 5)}{(x + 6) (x - 6)} (x + 6))$

Step 5

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 5.1

Cancel the common factor of $x + 6$ .

Tap for more steps...

Step 5.1.1

Cancel the common factor.

$\log_{7} (\frac{(x - 6) (x - 5)}{(x + 6) (x - 6)} (x + 6))$

Step 5.1.2

Rewrite the expression.

$\log_{7} (\frac{(x - 6) (x - 5)}{x - 6})$

Step 5.2

Cancel the common factor of $x - 6$ .

Tap for more steps...

Step 5.2.1

Cancel the common factor.

$\log_{7} (\frac{(x - 6) (x - 5)}{x - 6})$

Step 5.2.2

Divide $x - 5$ by $1$ .

$\log_{7} (x - 5)$