Evaluate log base 8 of square root of 2-2 log of 10+8

$\log_{8} (\sqrt{2}) - 2 \log (10) + 8$

Simplify each term.

Logarithm base $10$ of $10$ is $1$ .

$\log_{8} (\sqrt{2}) - 2 \cdot 1 + 8$

Multiply $- 2$ by $1$ .

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

Add $- 2$ and $8$ .

$\log_{8} (\sqrt{2}) + 6$

Rewrite $\log_{8} (\sqrt{2})$ 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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} + 6$

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

$\frac{\log (\sqrt{2})}{\log (8)} + 6$

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (\sqrt{2})}{\log (8)} + 6$

Decimal Form:

$6.1\overline{6}$