Precalculus Examples

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

Step 1

Rewrite $\frac{c}{64}$ as ${(\frac{1}{8})}^{2} c$ .

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Step 1.1

Factor the perfect power $1^{2}$ out of $c$ .

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

Step 1.2

Factor the perfect power $8^{2}$ out of $64$ .

$\log_{4} (\sqrt{\frac{1^{2} c}{8^{2} \cdot 1}})$

Step 1.3

Rearrange the fraction $\frac{1^{2} c}{8^{2} \cdot 1}$ .

$\log_{4} (\sqrt{{(\frac{1}{8})}^{2} c})$

Step 2

Pull terms out from under the radical.

$\log_{4} (\frac{1}{8} \sqrt{c})$

Step 3

Combine $\frac{1}{8}$ and $\sqrt{c}$ .

$\log_{4} (\frac{\sqrt{c}}{8})$

Step 4

Rewrite $\log_{4} (\frac{\sqrt{c}}{8})$ using the change of base formula.

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Step 4.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 4.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (\frac{\sqrt{c}}{8})}{\log (4)}$