Evaluate log base 8 of square root of 8

$\log_{8} (\sqrt{8})$

Step 1

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$\log_{8} (\sqrt{4 (2)})$

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

$\log_{8} (\sqrt{2^{2} \cdot 2})$

Step 2

Pull terms out from under the radical.

$\log_{8} (2 \sqrt{2})$

Step 3

Rewrite $\log_{8} (2 \sqrt{2})$ 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 $b = 10$ .

$\frac{\log (2 \sqrt{2})}{\log (8)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (2 \sqrt{2})}{\log (8)}$

Decimal Form:

$0.5$