Precalculus Examples

$\log_{6} (\sqrt[3]{\frac{1}{6}})$

Step 1

Rewrite $\sqrt[3]{\frac{1}{6}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{6}}$ .

$\log_{6} (\frac{\sqrt[3]{1}}{\sqrt[3]{6}})$

Step 2

Any root of $1$ is $1$ .

$\log_{6} (\frac{1}{\sqrt[3]{6}})$

Step 3

Multiply $\frac{1}{\sqrt[3]{6}}$ by $\frac{{\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{2}}$ .

$\log_{6} (\frac{1}{\sqrt[3]{6}} \cdot \frac{{\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{2}})$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{1}{\sqrt[3]{6}}$ by $\frac{{\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{2}}$ .

$\log_{6} (\frac{{\sqrt[3]{6}}^{2}}{\sqrt[3]{6} {\sqrt[3]{6}}^{2}})$

Step 4.2

Raise $\sqrt[3]{6}$ to the power of $1$ .

$\log_{6} (\frac{{\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{1} {\sqrt[3]{6}}^{2}})$

Step 4.3

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

$\log_{6} (\frac{{\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{1 + 2}})$

Step 4.4

Add $1$ and $2$ .

$\log_{6} (\frac{{\sqrt[3]{6}}^{2}}{{\sqrt[3]{6}}^{3}})$

Step 4.5

Rewrite ${\sqrt[3]{6}}^{3}$ as $6$ .

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Step 4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{6}$ as $6^{\frac{1}{3}}$ .

$\log_{6} (\frac{{\sqrt[3]{6}}^{2}}{{(6^{\frac{1}{3}})}^{3}})$

Step 4.5.2

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

$\log_{6} (\frac{{\sqrt[3]{6}}^{2}}{6^{\frac{1}{3} \cdot 3}})$

Step 4.5.3

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

$\log_{6} (\frac{{\sqrt[3]{6}}^{2}}{6^{\frac{3}{3}}})$

Step 4.5.4

Cancel the common factor of $3$ .

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Step 4.5.4.1

Cancel the common factor.

$\log_{6} (\frac{{\sqrt[3]{6}}^{2}}{6^{\frac{3}{3}}})$

Step 4.5.4.2

Rewrite the expression.

$\log_{6} (\frac{{\sqrt[3]{6}}^{2}}{6^{1}})$

Step 4.5.5

Evaluate the exponent.

$\log_{6} (\frac{{\sqrt[3]{6}}^{2}}{6})$

Step 5

Rewrite $\frac{{\sqrt[3]{6}}^{2}}{6}$ as ${\sqrt[3]{6}}^{2} \cdot 6^{- 1}$ .

$\log_{6} ({\sqrt[3]{6}}^{2} \cdot 6^{- 1})$

Step 6

Rewrite $\log_{6} ({\sqrt[3]{6}}^{2} \cdot 6^{- 1})$ as $\log_{6} ({\sqrt[3]{6}}^{2}) + \log_{6} (6^{- 1})$ .

$\log_{6} ({\sqrt[3]{6}}^{2}) + \log_{6} (6^{- 1})$

Step 7

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

$\log_{6} ({\sqrt[3]{6}}^{2}) - \log_{6} (6)$

Step 8

Logarithm base $6$ of $6$ is $1$ .

$\log_{6} ({\sqrt[3]{6}}^{2}) - 1 \cdot 1$

Step 9

Multiply $- 1$ by $1$ .

$\log_{6} ({\sqrt[3]{6}}^{2}) - 1$

Step 10

Expand $\log_{6} ({\sqrt[3]{6}}^{2})$ by moving $2$ outside the logarithm.

$2 \log_{6} (\sqrt[3]{6}) - 1$

Step 11

The result can be shown in multiple forms.

Exact Form:

$2 \log_{6} (\sqrt[3]{6}) - 1$

Decimal Form:

$- 0.\overline{3}$