Precalculus Examples

$\log (x) + \log (x^{2} - 9) - \log (7 - \log (x - 3))$

Step 1

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

$\log (x (x^{2} - 9)) - \log (7 - \log (x - 3))$

Step 2

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

$\log (\frac{x (x^{2} - 9)}{7 - \log (x - 3)})$

Step 3

Simplify the numerator.

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Step 3.1

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

$\log (\frac{x (x^{2} - 3^{2})}{7 - \log (x - 3)})$

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 $b = 3$ .

$\log (\frac{x (x + 3) (x - 3)}{7 - \log (x - 3)})$