Precalculus Examples

$\log_{4} (\frac{1}{\sqrt[3]{16}})$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $16$ as $2^{3} \cdot 2$ .

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Step 1.1.1

Factor $8$ out of $16$ .

$\log_{4} (\frac{1}{\sqrt[3]{8 (2)}})$

Step 1.1.2

Rewrite $8$ as $2^{3}$ .

$\log_{4} (\frac{1}{\sqrt[3]{2^{3} \cdot 2}})$

Step 1.2

Pull terms out from under the radical.

$\log_{4} (\frac{1}{2 \sqrt[3]{2}})$

Step 2

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

$\log_{4} (\frac{1}{2 \sqrt[3]{2}} \cdot \frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}})$

Step 3

Combine and simplify the denominator.

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Step 3.1

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

$\log_{4} (\frac{{\sqrt[3]{2}}^{2}}{2 \sqrt[3]{2} {\sqrt[3]{2}}^{2}})$

Step 3.2

Move $\sqrt[3]{2}$ .

$\log_{4} (\frac{{\sqrt[3]{2}}^{2}}{2 (\sqrt[3]{2} {\sqrt[3]{2}}^{2})})$

Step 3.3

Raise $\sqrt[3]{2}$ to the power of $1$ .

$\log_{4} (\frac{{\sqrt[3]{2}}^{2}}{2 ({\sqrt[3]{2}}^{1} {\sqrt[3]{2}}^{2})})$

Step 3.4

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

$\log_{4} (\frac{{\sqrt[3]{2}}^{2}}{2 {\sqrt[3]{2}}^{1 + 2}})$

Step 3.5

Add $1$ and $2$ .

$\log_{4} (\frac{{\sqrt[3]{2}}^{2}}{2 {\sqrt[3]{2}}^{3}})$

Step 3.6

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

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Step 3.6.1

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

$\log_{4} (\frac{{\sqrt[3]{2}}^{2}}{2 {(2^{\frac{1}{3}})}^{3}})$

Step 3.6.2

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

$\log_{4} (\frac{{\sqrt[3]{2}}^{2}}{2 \cdot 2^{\frac{1}{3} \cdot 3}})$

Step 3.6.3

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

$\log_{4} (\frac{{\sqrt[3]{2}}^{2}}{2 \cdot 2^{\frac{3}{3}}})$

Step 3.6.4

Cancel the common factor of $3$ .

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Step 3.6.4.1

Cancel the common factor.

$\log_{4} (\frac{{\sqrt[3]{2}}^{2}}{2 \cdot 2^{\frac{3}{3}}})$

Step 3.6.4.2

Rewrite the expression.

$\log_{4} (\frac{{\sqrt[3]{2}}^{2}}{2 \cdot 2^{1}})$

Step 3.6.5

Evaluate the exponent.

$\log_{4} (\frac{{\sqrt[3]{2}}^{2}}{2 \cdot 2})$

Step 4

Simplify the numerator.

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Step 4.1

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

$\log_{4} (\frac{\sqrt[3]{2^{2}}}{2 \cdot 2})$

Step 4.2

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

$\log_{4} (\frac{\sqrt[3]{4}}{2 \cdot 2})$

Step 5

Multiply $2$ by $2$ .

$\log_{4} (\frac{\sqrt[3]{4}}{4})$

Step 6

Rewrite $\frac{\sqrt[3]{4}}{4}$ as $\sqrt[3]{4} \cdot 4^{- 1}$ .

$\log_{4} (\sqrt[3]{4} \cdot 4^{- 1})$

Step 7

Rewrite $\log_{4} (\sqrt[3]{4} \cdot 4^{- 1})$ as $\log_{4} (\sqrt[3]{4}) + \log_{4} (4^{- 1})$ .

$\log_{4} (\sqrt[3]{4}) + \log_{4} (4^{- 1})$

Step 8

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

$\log_{4} (\sqrt[3]{4}) - \log_{4} (4)$

Step 9

Logarithm base $4$ of $4$ is $1$ .

$\log_{4} (\sqrt[3]{4}) - 1 \cdot 1$

Step 10

Multiply $- 1$ by $1$ .

$\log_{4} (\sqrt[3]{4}) - 1$

Step 11

Rewrite $\log_{4} (\sqrt[3]{4})$ 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 $b = 10$ .

$\frac{\log (\sqrt[3]{4})}{\log (4)} - 1$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (\sqrt[3]{4})}{\log (4)} - 1$

Decimal Form:

$- 0.\overline{6}$