Precalculus Examples

$\log_{4} (x^{2} - 9) - 5 \log_{4} (x + 3)$

Step 1

Simplify $- 5 \log_{4} (x + 3)$ by moving $5$ inside the logarithm.

$\log_{4} (x^{2} - 9) - \log_{4} ({(x + 3)}^{5})$

Step 2

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

$\log_{4} (\frac{x^{2} - 9}{{(x + 3)}^{5}})$

Step 3

Simplify the numerator.

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Step 3.1

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

$\log_{4} (\frac{x^{2} - 3^{2}}{{(x + 3)}^{5}})$

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 $b = 3$ .

$\log_{4} (\frac{(x + 3) (x - 3)}{{(x + 3)}^{5}})$

Step 4

Cancel the common factors.

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Step 4.1

Factor $x + 3$ out of ${(x + 3)}^{5}$ .

$\log_{4} (\frac{(x + 3) (x - 3)}{(x + 3) {(x + 3)}^{4}})$

Step 4.2

Cancel the common factor.

$\log_{4} (\frac{(x + 3) (x - 3)}{(x + 3) {(x + 3)}^{4}})$

Step 4.3

Rewrite the expression.

$\log_{4} (\frac{x - 3}{{(x + 3)}^{4}})$