Precalculus Examples

$2 \log_{3} (\sqrt{27})$

Step 1

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

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Step 1.1

Factor $9$ out of $27$ .

$2 \log_{3} (\sqrt{9 (3)})$

Step 1.2

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

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

Step 2

Pull terms out from under the radical.

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

Step 3

Simplify $2 \log_{3} (3 \sqrt{3})$ by moving $2$ inside the logarithm.

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

Step 4

Simplify the expression.

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Step 4.1

Apply the product rule to $3 \sqrt{3}$ .

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

Step 4.2

Raise $3$ to the power of $2$ .

$\log_{3} (9 {\sqrt{3}}^{2})$

Step 5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\log_{3} (9 {(3^{\frac{1}{2}})}^{2})$

Step 5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log_{3} (9 \cdot 3^{\frac{1}{2} \cdot 2})$

Step 5.3

Combine $\frac{1}{2}$ and $2$ .

$\log_{3} (9 \cdot 3^{\frac{2}{2}})$

Step 5.4

Cancel the common factor of $2$ .

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Step 5.4.1

Cancel the common factor.

$\log_{3} (9 \cdot 3^{\frac{2}{2}})$

Step 5.4.2

Rewrite the expression.

$\log_{3} (9 \cdot 3^{1})$

Step 5.5

Evaluate the exponent.

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

Step 6

Simplify the expression.

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Step 6.1

Multiply $9$ by $3$ .

$\log_{3} (27)$

Step 6.2

Logarithm base $3$ of $27$ is $3$ .

$3$