Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Apply the product rule to $x \sqrt[3]{y}$ .

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

Step 1.3

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

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

Step 1.4

Simplify $- 3 \log (\frac{y}{x})$ by moving $3$ inside the logarithm.

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

Step 1.5

Apply the product rule to $\frac{y}{x}$ .

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

Step 2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

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

Step 3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

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

Step 4

Cancel the common factor of $x^{3}$ and $x^{2}$ .

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Step 4.1

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

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

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

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

Combine.

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

Step 7

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 7.1

Move $x^{3}$ .

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

Step 7.2

Multiply $x^{3}$ by $x$ .

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Step 7.2.1

Raise $x$ to the power of $1$ .

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

Step 7.2.2

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

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

Step 7.3

Add $3$ and $1$ .

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

Step 8

Cancel the common factor of $y^{2}$ and $y^{3}$ .

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Step 8.1

Factor $y^{2}$ out of $x^{4} y^{2}$ .

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

Step 8.2

Cancel the common factors.

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Step 8.2.1

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

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

Step 8.2.2

Cancel the common factor.

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

Step 8.2.3

Rewrite the expression.

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

Step 9

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

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

Step 10

Combine and simplify the denominator.

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

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

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

Step 10.5

Add $1$ and $2$ .

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

Step 10.6

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

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Step 10.6.1

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

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

Step 10.6.2

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

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

Step 10.6.3

Combine $\frac{2}{3}$ and $3$ .

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

Step 10.6.4

Multiply $2$ by $3$ .

$\log (\frac{x^{4} {\sqrt[3]{y^{2}}}^{2}}{y \cdot y^{\frac{6}{3}}})$

Step 10.6.5

Cancel the common factor of $6$ and $3$ .

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Step 10.6.5.1

Factor $3$ out of $6$ .

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

Step 10.6.5.2

Cancel the common factors.

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Step 10.6.5.2.1

Factor $3$ out of $3$ .

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

Step 10.6.5.2.2

Cancel the common factor.

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

Step 10.6.5.2.3

Rewrite the expression.

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

Step 10.6.5.2.4

Divide $2$ by $1$ .

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

Step 11

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 11.1

Multiply $y$ by $y^{2}$ .

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Step 11.1.1

Raise $y$ to the power of $1$ .

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

Step 11.1.2

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

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

Step 11.2

Add $1$ and $2$ .

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

Step 12

Simplify the numerator.

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Step 12.1

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

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

Step 12.2

Multiply the exponents in ${(y^{2})}^{2}$ .

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Step 12.2.1

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

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

Step 12.2.2

Multiply $2$ by $2$ .

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

Step 12.3

Factor out $y^{3}$ .

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

Step 12.4

Pull terms out from under the radical.

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

Step 13

Cancel the common factor of $y$ and $y^{3}$ .

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Step 13.1

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

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

Step 13.2

Cancel the common factors.

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Step 13.2.1

Factor $y$ out of $y^{3}$ .

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

Step 13.2.2

Cancel the common factor.

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

Step 13.2.3

Rewrite the expression.

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