Precalculus Examples

$\ln ({(\frac{x}{x^{2} - 16})}^{4})$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $16$ as $4^{2}$ .

$\ln ({(\frac{x}{x^{2} - 4^{2}})}^{4})$

Step 1.2

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 $b = 4$ .

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

Step 2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1

Apply the product rule to $\frac{x}{(x + 4) (x - 4)}$ .

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

Step 2.2

Apply the product rule to $(x + 4) (x - 4)$ .

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