Precalculus Examples

$\log_{4} (\sqrt[4]{64})$

Step 1

Rewrite $64$ as $2^{4} \cdot 4$ .

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Step 1.1

Factor $16$ out of $64$ .

$\log_{4} (\sqrt[4]{16 (4)})$

Step 1.2

Rewrite $16$ as $2^{4}$ .

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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Step 6.1

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 6.2

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

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.75$