Precalculus Examples

$\log (x^{2} - 36) - 8 \log (x + 6) + 6 \log (x)$

Step 1

Simplify each term.

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Step 1.1

Simplify $- 8 \log (x + 6)$ by moving $8$ inside the logarithm.

$\log (x^{2} - 36) - \log ({(x + 6)}^{8}) + 6 \log (x)$

Step 1.2

Simplify $6 \log (x)$ by moving $6$ inside the logarithm.

$\log (x^{2} - 36) - \log ({(x + 6)}^{8}) + \log (x^{6})$

Step 2

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

$\log (\frac{x^{2} - 36}{{(x + 6)}^{8}}) + \log (x^{6})$

Step 3

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

$\log (\frac{x^{2} - 36}{{(x + 6)}^{8}} x^{6})$

Step 4

Simplify the numerator.

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Step 4.1

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

$\log (\frac{x^{2} - 6^{2}}{{(x + 6)}^{8}} x^{6})$

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 $b = 6$ .

$\log (\frac{(x + 6) (x - 6)}{{(x + 6)}^{8}} x^{6})$

Step 5

Cancel the common factors.

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Step 5.1

Factor $x + 6$ out of ${(x + 6)}^{8}$ .

$\log (\frac{(x + 6) (x - 6)}{(x + 6) {(x + 6)}^{7}} x^{6})$

Step 5.2

Cancel the common factor.

$\log (\frac{(x + 6) (x - 6)}{(x + 6) {(x + 6)}^{7}} x^{6})$

Step 5.3

Rewrite the expression.

$\log (\frac{x - 6}{{(x + 6)}^{7}} x^{6})$

Step 6

Combine $\frac{x - 6}{{(x + 6)}^{7}}$ and $x^{6}$ .

$\log (\frac{(x - 6) x^{6}}{{(x + 6)}^{7}})$

Step 7

Reorder factors in $\log (\frac{(x - 6) x^{6}}{{(x + 6)}^{7}})$ .

$\log (\frac{x^{6} (x - 6)}{{(x + 6)}^{7}})$