Precalculus Examples

$2 \log (x) - \log (x^{2} - 9)$

Step 1

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

$\log (x^{2}) - \log (x^{2} - 9)$

Step 2

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

$\log (\frac{x^{2}}{x^{2} - 9})$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $9$ as $3^{2}$ .

$\log (\frac{x^{2}}{x^{2} - 3^{2}})$

Step 3.2

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 $b = 3$ .

$\log (\frac{x^{2}}{(x + 3) (x - 3)})$