Precalculus Examples

$5 \log_{4} (x^{2} - 1) - 4 \log_{4} (x + 3)$

Step 1

Simplify each term.

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Step 1.1

Simplify $5 \log_{4} (x^{2} - 1)$ by moving $5$ inside the logarithm.

$\log_{4} ({(x^{2} - 1)}^{5}) - 4 \log_{4} (x + 3)$

Step 1.2

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

$\log_{4} ({(x^{2} - 1)}^{5}) - \log_{4} ({(x + 3)}^{4})$

Step 2

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

$\log_{4} (\frac{{(x^{2} - 1)}^{5}}{{(x + 3)}^{4}})$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $1$ as $1^{2}$ .

$\log_{4} (\frac{{(x^{2} - 1^{2})}^{5}}{{(x + 3)}^{4}})$

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 $b = 1$ .

$\log_{4} (\frac{{((x + 1) (x - 1))}^{5}}{{(x + 3)}^{4}})$

Step 3.3

Apply the product rule to $(x + 1) (x - 1)$ .

$\log_{4} (\frac{{(x + 1)}^{5} {(x - 1)}^{5}}{{(x + 3)}^{4}})$