Precalculus Examples

$\log_{3} (\sqrt[3]{\frac{1}{9}})$

Step 1

Rewrite $\sqrt[3]{\frac{1}{9}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{9}}$ .

$\log_{3} (\frac{\sqrt[3]{1}}{\sqrt[3]{9}})$

Step 2

Any root of $1$ is $1$ .

$\log_{3} (\frac{1}{\sqrt[3]{9}})$

Step 3

Multiply $\frac{1}{\sqrt[3]{9}}$ by $\frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{2}}$ .

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

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{1}{\sqrt[3]{9}}$ by $\frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{2}}$ .

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

Step 4.2

Raise $\sqrt[3]{9}$ to the power of $1$ .

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

Step 4.3

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

$\log_{3} (\frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{1 + 2}})$

Step 4.4

Add $1$ and $2$ .

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

Step 4.5

Rewrite ${\sqrt[3]{9}}^{3}$ as $9$ .

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Step 4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{9}$ as $9^{\frac{1}{3}}$ .

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

Step 4.5.2

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

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

Step 4.5.3

Combine $\frac{1}{3}$ and $3$ .

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

Step 4.5.4

Cancel the common factor of $3$ .

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Step 4.5.4.1

Cancel the common factor.

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

Step 4.5.4.2

Rewrite the expression.

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

Step 4.5.5

Evaluate the exponent.

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

Step 5

Simplify the numerator.

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Step 5.1

Rewrite ${\sqrt[3]{9}}^{2}$ as $\sqrt[3]{9^{2}}$ .

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

Step 5.2

Raise $9$ to the power of $2$ .

$\log_{3} (\frac{\sqrt[3]{81}}{9})$

Step 5.3

Rewrite $81$ as $3^{3} \cdot 3$ .

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Step 5.3.1

Factor $27$ out of $81$ .

$\log_{3} (\frac{\sqrt[3]{27 (3)}}{9})$

Step 5.3.2

Rewrite $27$ as $3^{3}$ .

$\log_{3} (\frac{\sqrt[3]{3^{3} \cdot 3}}{9})$

Step 5.4

Pull terms out from under the radical.

$\log_{3} (\frac{3 \sqrt[3]{3}}{9})$

Step 6

Reduce the expression $\frac{3 \sqrt[3]{3}}{9}$ by cancelling the common factors.

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Step 6.1

Factor $3$ out of $3 \sqrt[3]{3}$ .

$\log_{3} (\frac{3 (\sqrt[3]{3})}{9})$

Step 6.2

Factor $3$ out of $9$ .

$\log_{3} (\frac{3 \sqrt[3]{3}}{3 \cdot 3})$

Step 6.3

Cancel the common factor.

$\log_{3} (\frac{3 \sqrt[3]{3}}{3 \cdot 3})$

Step 6.4

Rewrite the expression.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Multiply $- 1$ by $1$ .

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

Step 12

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

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Step 12.1

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 12.2

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

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

Step 13

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.\overline{6}$