Precalculus Examples

$\log_{4} (\sqrt[6]{4})$

Step 1

Rewrite $4$ as $2^{2}$ .

$\log_{4} (\sqrt[6]{2^{2}})$

Step 2

Rewrite $\sqrt[6]{2^{2}}$ as $\sqrt[3]{\sqrt{2^{2}}}$ .

$\log_{4} (\sqrt[3]{\sqrt{2^{2}}})$

Step 3

Pull terms out from under the radical, assuming positive real numbers.

$\log_{4} (\sqrt[3]{2})$

Step 4

Rewrite $\log_{4} (\sqrt[3]{2})$ 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 $b = 10$ .

$\frac{\log (\sqrt[3]{2})}{\log (4)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (\sqrt[3]{2})}{\log (4)}$

Decimal Form:

$0.1\overline{6}$