Evaluate log base 12 of square root of 12

$\log_{12} (\sqrt{12})$

Step 1

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$\log_{12} (\sqrt{4 (3)})$

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

$\log_{12} (\sqrt{2^{2} \cdot 3})$

Step 2

Pull terms out from under the radical.

$\log_{12} (2 \sqrt{3})$

Step 3

Rewrite $\log_{12} (2 \sqrt{3})$ 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 $b = 10$ .

$\frac{\log (2 \sqrt{3})}{\log (12)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (2 \sqrt{3})}{\log (12)}$

Decimal Form:

$0.5$