Precalculus Examples

$\log_{2} (\frac{16 x^{3}}{\sqrt[3]{32 y}})$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $32 y$ as $2^{3} \cdot (4 y)$ .

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Step 1.1.1

Factor $8$ out of $32$ .

$\log_{2} (\frac{16 x^{3}}{\sqrt[3]{8 (4) y}})$

Step 1.1.2

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

$\log_{2} (\frac{16 x^{3}}{\sqrt[3]{2^{3} \cdot 4 y}})$

Step 1.1.3

Add parentheses.

$\log_{2} (\frac{16 x^{3}}{\sqrt[3]{2^{3} \cdot (4 y)}})$

Step 1.2

Pull terms out from under the radical.

$\log_{2} (\frac{16 x^{3}}{2 \sqrt[3]{4 y}})$

Step 2

Cancel the common factor of $16$ and $2$ .

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Step 2.1

Factor $2$ out of $16 x^{3}$ .

$\log_{2} (\frac{2 (8 x^{3})}{2 \sqrt[3]{4 y}})$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $2$ out of $2 \sqrt[3]{4 y}$ .

$\log_{2} (\frac{2 (8 x^{3})}{2 (\sqrt[3]{4 y})})$

Step 2.2.2

Cancel the common factor.

$\log_{2} (\frac{2 (8 x^{3})}{2 \sqrt[3]{4 y}})$

Step 2.2.3

Rewrite the expression.

$\log_{2} (\frac{8 x^{3}}{\sqrt[3]{4 y}})$

Step 3

Multiply $\frac{8 x^{3}}{\sqrt[3]{4 y}}$ by $\frac{{\sqrt[3]{4 y}}^{2}}{{\sqrt[3]{4 y}}^{2}}$ .

$\log_{2} (\frac{8 x^{3}}{\sqrt[3]{4 y}} \cdot \frac{{\sqrt[3]{4 y}}^{2}}{{\sqrt[3]{4 y}}^{2}})$

Step 4

Simplify terms.

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Step 4.1

Combine and simplify the denominator.

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Step 4.1.1

Multiply $\frac{8 x^{3}}{\sqrt[3]{4 y}}$ by $\frac{{\sqrt[3]{4 y}}^{2}}{{\sqrt[3]{4 y}}^{2}}$ .

$\log_{2} (\frac{8 x^{3} {\sqrt[3]{4 y}}^{2}}{\sqrt[3]{4 y} {\sqrt[3]{4 y}}^{2}})$

Step 4.1.2

Raise $\sqrt[3]{4 y}$ to the power of $1$ .

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

Step 4.1.3

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

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

Step 4.1.4

Add $1$ and $2$ .

$\log_{2} (\frac{8 x^{3} {\sqrt[3]{4 y}}^{2}}{{\sqrt[3]{4 y}}^{3}})$

Step 4.1.5

Rewrite ${\sqrt[3]{4 y}}^{3}$ as $4 y$ .

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Step 4.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4 y}$ as ${(4 y)}^{\frac{1}{3}}$ .

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

Step 4.1.5.2

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

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

Step 4.1.5.3

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

$\log_{2} (\frac{8 x^{3} {\sqrt[3]{4 y}}^{2}}{{(4 y)}^{\frac{3}{3}}})$

Step 4.1.5.4

Cancel the common factor of $3$ .

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Step 4.1.5.4.1

Cancel the common factor.

$\log_{2} (\frac{8 x^{3} {\sqrt[3]{4 y}}^{2}}{{(4 y)}^{\frac{3}{3}}})$

Step 4.1.5.4.2

Rewrite the expression.

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

Step 4.1.5.5

Simplify.

$\log_{2} (\frac{8 x^{3} {\sqrt[3]{4 y}}^{2}}{4 y})$

Step 4.2

Cancel the common factor of $8$ and $4$ .

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Step 4.2.1

Factor $4$ out of $8 x^{3} {\sqrt[3]{4 y}}^{2}$ .

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

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $4$ out of $4 y$ .

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

Step 4.2.2.2

Cancel the common factor.

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

Step 4.2.2.3

Rewrite the expression.

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

Step 5

Simplify the numerator.

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Step 5.1

Rewrite ${\sqrt[3]{4 y}}^{2}$ as $\sqrt[3]{{(4 y)}^{2}}$ .

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

Step 5.2

Apply the product rule to $4 y$ .

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

Step 5.3

Raise $4$ to the power of $2$ .

$\log_{2} (\frac{2 x^{3} \sqrt[3]{16 y^{2}}}{y})$

Step 5.4

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

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Step 5.4.1

Factor $8$ out of $16$ .

$\log_{2} (\frac{2 x^{3} \sqrt[3]{8 (2) y^{2}}}{y})$

Step 5.4.2

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

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

Step 5.4.3

Add parentheses.

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

Step 5.5

Pull terms out from under the radical.

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

Step 5.6

Multiply $2$ by $2$ .

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

Step 6

Rewrite $\log_{2} (\frac{4 x^{3} \sqrt[3]{2 y^{2}}}{y})$ using the change of base formula.

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Step 6.1

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 $0$ .

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

Step 6.2

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

$\frac{\log (\frac{4 x^{3} \sqrt[3]{2 y^{2}}}{y})}{\log (2)}$