Precalculus Examples

$\log_{2} (\sqrt[3]{\frac{16 (x + 1)}{(x^{2} + 9) (x + 3)}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{\frac{16 (x + 1)}{(x^{2} + 9) (x + 3)}}$ as ${(\frac{16 (x + 1)}{(x^{2} + 9) (x + 3)})}^{\frac{1}{3}}$ .

$\log_{2} ({(\frac{16 (x + 1)}{(x^{2} + 9) (x + 3)})}^{\frac{1}{3}})$

Step 2

Expand $\log_{2} ({(\frac{16 (x + 1)}{(x^{2} + 9) (x + 3)})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \log_{2} (\frac{16 (x + 1)}{(x^{2} + 9) (x + 3)})$

Step 3

Rewrite $\log_{2} (\frac{16 (x + 1)}{(x^{2} + 9) (x + 3)})$ as $\log_{2} (16 (x + 1)) - \log_{2} ((x^{2} + 9) (x + 3))$ .

$\frac{1}{3} (\log_{2} (16 (x + 1)) - \log_{2} ((x^{2} + 9) (x + 3)))$

Step 4

Rewrite $\log_{2} (16 (x + 1))$ as $\log_{2} (16) + \log_{2} (x + 1)$ .

$\frac{1}{3} (\log_{2} (16) + \log_{2} (x + 1) - \log_{2} ((x^{2} + 9) (x + 3)))$

Step 5

Logarithm base $2$ of $16$ is $4$ .

$\frac{1}{3} (4 + \log_{2} (x + 1) - \log_{2} ((x^{2} + 9) (x + 3)))$

Step 6

Rewrite $\log_{2} ((x^{2} + 9) (x + 3))$ as $\log_{2} (x^{2} + 9) + \log_{2} (x + 3)$ .

$\frac{1}{3} (4 + \log_{2} (x + 1) - (\log_{2} (x^{2} + 9) + \log_{2} (x + 3)))$

Step 7

Apply the distributive property.

$\frac{1}{3} (4 + \log_{2} (x + 1) - \log_{2} (x^{2} + 9) - \log_{2} (x + 3))$

Step 8

Apply the distributive property.

$\frac{1}{3} \cdot 4 + \frac{1}{3} \log_{2} (x + 1) + \frac{1}{3} (- \log_{2} (x^{2} + 9)) + \frac{1}{3} (- \log_{2} (x + 3))$

Step 9

Simplify.

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Step 9.1

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

$\frac{4}{3} + \frac{1}{3} \log_{2} (x + 1) + \frac{1}{3} (- \log_{2} (x^{2} + 9)) + \frac{1}{3} (- \log_{2} (x + 3))$

Step 9.2

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

$\frac{4}{3} + \frac{\log_{2} (x + 1)}{3} + \frac{1}{3} (- \log_{2} (x^{2} + 9)) + \frac{1}{3} (- \log_{2} (x + 3))$

Step 9.3

Combine $\frac{1}{3}$ and $\log_{2} (x^{2} + 9)$ .

$\frac{4}{3} + \frac{\log_{2} (x + 1)}{3} - \frac{\log_{2} (x^{2} + 9)}{3} + \frac{1}{3} (- \log_{2} (x + 3))$

Step 9.4

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

$\frac{4}{3} + \frac{\log_{2} (x + 1)}{3} - \frac{\log_{2} (x^{2} + 9)}{3} - \frac{\log_{2} (x + 3)}{3}$

Step 10

Combine the numerators over the common denominator.

$\frac{4 + \log_{2} (x + 1)}{3} - \frac{\log_{2} (x^{2} + 9)}{3} - \frac{\log_{2} (x + 3)}{3}$

Step 11

Combine the numerators over the common denominator.

$\frac{4 + \log_{2} (x + 1) - \log_{2} (x^{2} + 9)}{3} - \frac{\log_{2} (x + 3)}{3}$

Step 12

Combine the numerators over the common denominator.

$\frac{4 + \log_{2} (x + 1) - \log_{2} (x^{2} + 9) - \log_{2} (x + 3)}{3}$