Precalculus Examples

$\log_{6} (\frac{1}{\sqrt[5]{36}})$

Step 1

Multiply $\frac{1}{\sqrt[5]{36}}$ by $\frac{{\sqrt[5]{36}}^{4}}{{\sqrt[5]{36}}^{4}}$ .

$\log_{6} (\frac{1}{\sqrt[5]{36}} \cdot \frac{{\sqrt[5]{36}}^{4}}{{\sqrt[5]{36}}^{4}})$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{1}{\sqrt[5]{36}}$ by $\frac{{\sqrt[5]{36}}^{4}}{{\sqrt[5]{36}}^{4}}$ .

$\log_{6} (\frac{{\sqrt[5]{36}}^{4}}{\sqrt[5]{36} {\sqrt[5]{36}}^{4}})$

Step 2.2

Raise $\sqrt[5]{36}$ to the power of $1$ .

$\log_{6} (\frac{{\sqrt[5]{36}}^{4}}{{\sqrt[5]{36}}^{1} {\sqrt[5]{36}}^{4}})$

Step 2.3

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

$\log_{6} (\frac{{\sqrt[5]{36}}^{4}}{{\sqrt[5]{36}}^{1 + 4}})$

Step 2.4

Add $1$ and $4$ .

$\log_{6} (\frac{{\sqrt[5]{36}}^{4}}{{\sqrt[5]{36}}^{5}})$

Step 2.5

Rewrite ${\sqrt[5]{36}}^{5}$ as $36$ .

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{36}$ as $36^{\frac{1}{5}}$ .

$\log_{6} (\frac{{\sqrt[5]{36}}^{4}}{{(36^{\frac{1}{5}})}^{5}})$

Step 2.5.2

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

$\log_{6} (\frac{{\sqrt[5]{36}}^{4}}{36^{\frac{1}{5} \cdot 5}})$

Step 2.5.3

Combine $\frac{1}{5}$ and $5$ .

$\log_{6} (\frac{{\sqrt[5]{36}}^{4}}{36^{\frac{5}{5}}})$

Step 2.5.4

Cancel the common factor of $5$ .

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Step 2.5.4.1

Cancel the common factor.

$\log_{6} (\frac{{\sqrt[5]{36}}^{4}}{36^{\frac{5}{5}}})$

Step 2.5.4.2

Rewrite the expression.

$\log_{6} (\frac{{\sqrt[5]{36}}^{4}}{36^{1}})$

Step 2.5.5

Evaluate the exponent.

$\log_{6} (\frac{{\sqrt[5]{36}}^{4}}{36})$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite ${\sqrt[5]{36}}^{4}$ as $\sqrt[5]{36^{4}}$ .

$\log_{6} (\frac{\sqrt[5]{36^{4}}}{36})$

Step 3.2

Raise $36$ to the power of $4$ .

$\log_{6} (\frac{\sqrt[5]{1679616}}{36})$

Step 3.3

Rewrite $1679616$ as $6^{5} \cdot 216$ .

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Step 3.3.1

Factor $7776$ out of $1679616$ .

$\log_{6} (\frac{\sqrt[5]{7776 (216)}}{36})$

Step 3.3.2

Rewrite $7776$ as $6^{5}$ .

$\log_{6} (\frac{\sqrt[5]{6^{5} \cdot 216}}{36})$

Step 3.4

Pull terms out from under the radical.

$\log_{6} (\frac{6 \sqrt[5]{216}}{36})$

Step 4

Reduce the expression $\frac{6 \sqrt[5]{216}}{36}$ by cancelling the common factors.

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Step 4.1

Factor $6$ out of $6 \sqrt[5]{216}$ .

$\log_{6} (\frac{6 (\sqrt[5]{216})}{36})$

Step 4.2

Factor $6$ out of $36$ .

$\log_{6} (\frac{6 \sqrt[5]{216}}{6 \cdot 6})$

Step 4.3

Cancel the common factor.

$\log_{6} (\frac{6 \sqrt[5]{216}}{6 \cdot 6})$

Step 4.4

Rewrite the expression.

$\log_{6} (\frac{\sqrt[5]{216}}{6})$

Step 5

Rewrite $\frac{\sqrt[5]{216}}{6}$ as $\sqrt[5]{216} \cdot 6^{- 1}$ .

$\log_{6} (\sqrt[5]{216} \cdot 6^{- 1})$

Step 6

Rewrite $\log_{6} (\sqrt[5]{216} \cdot 6^{- 1})$ as $\log_{6} (\sqrt[5]{216}) + \log_{6} (6^{- 1})$ .

$\log_{6} (\sqrt[5]{216}) + \log_{6} (6^{- 1})$

Step 7

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

$\log_{6} (\sqrt[5]{216}) - \log_{6} (6)$

Step 8

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

$\log_{6} (\sqrt[5]{216}) - 1 \cdot 1$

Step 9

Multiply $- 1$ by $1$ .

$\log_{6} (\sqrt[5]{216}) - 1$

Step 10

Rewrite $\log_{6} (\sqrt[5]{216})$ using the change of base formula.

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Step 10.1

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 10.2

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

$\frac{\log (\sqrt[5]{216})}{\log (6)} - 1$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (\sqrt[5]{216})}{\log (6)} - 1$

Decimal Form:

$- 0.4$