Algebra Examples

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

Step 1

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

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Step 1.1

Factor $9$ out of $27$ .

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

Step 1.2

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

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

Step 2

Pull terms out from under the radical.

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

Step 3

Rewrite $\log_{3} (3 \sqrt{3})$ using the change of base formula.

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Step 3.1

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 3.2

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

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.5$