Precalculus Examples

$\log_{2} (\sqrt[4]{\frac{2 + x}{8}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{\frac{2 + x}{8}}$ as ${(\frac{2 + x}{8})}^{\frac{1}{4}}$ .

$\log_{2} ({(\frac{2 + x}{8})}^{\frac{1}{4}})$

Step 2

Expand $\log_{2} ({(\frac{2 + x}{8})}^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

$\frac{1}{4} \log_{2} (\frac{2 + x}{8})$

Step 3

Rewrite $\log_{2} (\frac{2 + x}{8})$ as $\log_{2} (2 + x) - \log_{2} (8)$ .

$\frac{1}{4} (\log_{2} (2 + x) - \log_{2} (8))$

Step 4

Logarithm base $2$ of $8$ is $3$ .

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

Step 5

Multiply $- 1$ by $3$ .

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

Step 6

Apply the distributive property.

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

Step 7

Combine $\frac{1}{4}$ and $\log_{2} (2 + x)$ .

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

Step 8

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

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

Step 9

Move the negative in front of the fraction.

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

Step 10

Combine the numerators over the common denominator.

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