Precalculus Examples

$\log (\sqrt[3]{\frac{x + 2}{x^{4} (x^{2} + 4)}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{\frac{x + 2}{x^{4} (x^{2} + 4)}}$ as ${(\frac{x + 2}{x^{4} (x^{2} + 4)})}^{\frac{1}{3}}$ .

$\log ({(\frac{x + 2}{x^{4} (x^{2} + 4)})}^{\frac{1}{3}})$

Step 2

Expand $\log ({(\frac{x + 2}{x^{4} (x^{2} + 4)})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \log (\frac{x + 2}{x^{4} (x^{2} + 4)})$

Step 3

Rewrite $\log (\frac{x + 2}{x^{4} (x^{2} + 4)})$ as $\log (x + 2) - \log (x^{4} (x^{2} + 4))$ .

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

Step 4

Rewrite $\log (x^{4} (x^{2} + 4))$ as $\log (x^{4}) + \log (x^{2} + 4)$ .

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

Step 5

Expand $\log (x^{4})$ by moving $4$ outside the logarithm.

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

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $4$ by $- 1$ .

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

Step 7

Apply the distributive property.

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

Step 8

Simplify.

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Step 8.1

Combine $\frac{1}{3}$ and $\log (x + 2)$ .

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

Step 8.2

Multiply $\frac{1}{3} (- 4 \log (x))$ .

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Step 8.2.1

Combine $- 4$ and $\frac{1}{3}$ .

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

Step 8.2.2

Combine $\frac{- 4}{3}$ and $\log (x)$ .

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

Step 8.3

Combine $\frac{1}{3}$ and $\log (x^{2} + 4)$ .

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

Step 9

Move the negative in front of the fraction.

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

Step 10

Combine the numerators over the common denominator.

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

Step 11

Combine the numerators over the common denominator.

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