Evaluate log base 6 of cube root of 1/6

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

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.

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

Simplify each term.

Rewrite $\log_{6} (\sqrt[3]{6})$ using the change of base formula.

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 $0$ .

$2 \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} - 1$

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

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

Multiply $2 \frac{\log (\sqrt[3]{6})}{\log (6)}$ .

Combine $2$ and $\frac{\log (\sqrt[3]{6})}{\log (6)}$ .

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

Simplify $2 \log (\sqrt[3]{6})$ by moving $2$ inside the logarithm.

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

Simplify the numerator.

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

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

Raise $6$ to the power of $2$ .

$\frac{\log (\sqrt[3]{36})}{\log (6)} - 1$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (\sqrt[3]{36})}{\log (6)} - 1$

Decimal Form:

$- 0.\overline{3}$

Evaluate log base 6 of cube root of 1/6

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