Precalculus Examples

$\log_{3} (\sqrt{\frac{27}{81}})$

Step 1

Cancel the common factor of $27$ and $81$ .

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Step 1.1

Factor $27$ out of $27$ .

$\log_{3} (\sqrt{\frac{27 (1)}{81}})$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $27$ out of $81$ .

$\log_{3} (\sqrt{\frac{27 \cdot 1}{27 \cdot 3}})$

Step 1.2.2

Cancel the common factor.

$\log_{3} (\sqrt{\frac{27 \cdot 1}{27 \cdot 3}})$

Step 1.2.3

Rewrite the expression.

$\log_{3} (\sqrt{\frac{1}{3}})$

Step 2

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$\log_{3} (\frac{\sqrt{1}}{\sqrt{3}})$

Step 3

Any root of $1$ is $1$ .

$\log_{3} (\frac{1}{\sqrt{3}})$

Step 4

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\log_{3} (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 5.2

Raise $\sqrt{3}$ to the power of $1$ .

$\log_{3} (\frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}})$

Step 5.3

Raise $\sqrt{3}$ to the power of $1$ .

$\log_{3} (\frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}})$

Step 5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log_{3} (\frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}})$

Step 5.5

Add $1$ and $1$ .

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

Step 5.6

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

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Step 5.6.1

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

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

Step 5.6.2

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

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

Step 5.6.3

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

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

Step 5.6.4

Cancel the common factor of $2$ .

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Step 5.6.4.1

Cancel the common factor.

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

Step 5.6.4.2

Rewrite the expression.

$\log_{3} (\frac{\sqrt{3}}{3^{1}})$

Step 5.6.5

Evaluate the exponent.

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

Step 6

Rewrite $\frac{\sqrt{3}}{3}$ as $\sqrt{3} \cdot 3^{- 1}$ .

$\log_{3} (\sqrt{3} \cdot 3^{- 1})$

Step 7

Rewrite $\log_{3} (\sqrt{3} \cdot 3^{- 1})$ as $\log_{3} (\sqrt{3}) + \log_{3} (3^{- 1})$ .

$\log_{3} (\sqrt{3}) + \log_{3} (3^{- 1})$

Step 8

Use logarithm rules to move $- 1$ out of the exponent.

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

Step 9

Logarithm base $3$ of $3$ is $1$ .

$\log_{3} (\sqrt{3}) - 1 \cdot 1$

Step 10

Multiply $- 1$ by $1$ .

$\log_{3} (\sqrt{3}) - 1$

Step 11

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

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Step 11.1

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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} - 1$

Step 11.2

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

$\frac{\log (\sqrt{3})}{\log (3)} - 1$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (\sqrt{3})}{\log (3)} - 1$

Decimal Form:

$- 0.5$