Evaluate log base 4 of fourth root of 4

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

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

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

Rewrite $\sqrt[4]{2^{2}}$ as $\sqrt{\sqrt{2^{2}}}$ .

$\log_{4} (\sqrt{\sqrt{2^{2}}})$

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

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

Rewrite $\log_{4} (\sqrt{2})$ using the change of base formula.

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 $0$ .

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

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

$\frac{\log (\sqrt{2})}{\log (4)}$

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (\sqrt{2})}{\log (4)}$

Decimal Form:

$0.25$