Precalculus Examples

$\log_{2} (\sqrt{\frac{x}{4}})$

Step 1

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

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

Step 2

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

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

Step 3

Rewrite $\log_{2} (\frac{x}{4})$ as $\log_{2} (x) - \log_{2} (4)$ .

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

Step 4

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

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

Step 5

Multiply $- 1$ by $2$ .

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

Step 6

Apply the distributive property.

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

Step 7

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

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

Step 8

Cancel the common factor of $2$ .

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Step 8.1

Factor $2$ out of $- 2$ .

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

Step 8.2

Cancel the common factor.

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

Step 8.3

Rewrite the expression.

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