Precalculus Examples

$\log_{4} (\sqrt{\frac{x}{64}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{x}{64}}$ as ${(\frac{x}{64})}^{\frac{1}{2}}$ .

$\log_{4} ({(\frac{x}{64})}^{\frac{1}{2}})$

Step 2

Expand $\log_{4} ({(\frac{x}{64})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{4} (\frac{x}{64})$

Step 3

Rewrite $\log_{4} (\frac{x}{64})$ as $\log_{4} (x) - \log_{4} (64)$ .

$\frac{1}{2} (\log_{4} (x) - \log_{4} (64))$

Step 4

Logarithm base $4$ of $64$ is $3$ .

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

Step 5

Multiply $- 1$ by $3$ .

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

Step 6

Apply the distributive property.

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

Step 7

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

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

Step 8

Combine $\frac{1}{2}$ and $- 3$ .

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

Step 9

Move the negative in front of the fraction.

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