Precalculus Examples

$\log (x + 19) + 2 \log (x - 19) - \log (x^{2} - 361)$

Step 1

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

$2 \log (x - 19) + \log (\frac{x + 19}{x^{2} - 361})$

Step 2

Simplify each term.

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Step 2.1

Simplify $2 \log (x - 19)$ by moving $2$ inside the logarithm.

$\log ({(x - 19)}^{2}) + \log (\frac{x + 19}{x^{2} - 361})$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Rewrite $361$ as $19^{2}$ .

$\log ({(x - 19)}^{2}) + \log (\frac{x + 19}{x^{2} - 19^{2}})$

Step 2.2.2

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 $b = 19$ .

$\log ({(x - 19)}^{2}) + \log (\frac{x + 19}{(x + 19) (x - 19)})$

Step 2.3

Cancel the common factor of $x + 19$ .

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Step 2.3.1

Cancel the common factor.

$\log ({(x - 19)}^{2}) + \log (\frac{x + 19}{(x + 19) (x - 19)})$

Step 2.3.2

Rewrite the expression.

$\log ({(x - 19)}^{2}) + \log (\frac{1}{x - 19})$

Step 3

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

$\log ({(x - 19)}^{2} \frac{1}{x - 19})$

Step 4

Cancel the common factor of $x - 19$ .

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Step 4.1

Factor $x - 19$ out of ${(x - 19)}^{2}$ .

$\log ((x - 19) (x - 19) \frac{1}{x - 19})$

Step 4.2

Cancel the common factor.

$\log ((x - 19) (x - 19) \frac{1}{x - 19})$

Step 4.3

Rewrite the expression.

$\log (x - 19)$