Evaluate log base 3 of square root of 27

$\log_{3} (\sqrt{27})$

Rewrite $27$ as $3^{2} \cdot 3$ .

Factor $9$ out of $27$ .

$\log_{3} (\sqrt{9 (3)})$

Rewrite $9$ as $3^{2}$ .

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

Pull terms out from under the radical.

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

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

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.5$