Precalculus Examples

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

Step 1

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

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

Step 2

Simplify each term.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

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

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Add $1$ and $2$ .

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

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

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

Step 3

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

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

Step 4

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

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

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

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

Step 6

Combine and simplify the denominator.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

Add $1$ and $2$ .

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

Step 6.5

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

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Step 6.5.1

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

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

Step 6.5.2

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

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

Step 6.5.3

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

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

Step 6.5.4

Multiply $2$ by $3$ .

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

Step 6.5.5

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

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Step 6.5.5.1

Factor $3$ out of $6$ .

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

Step 6.5.5.2

Cancel the common factors.

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Step 6.5.5.2.1

Factor $3$ out of $3$ .

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

Step 6.5.5.2.2

Cancel the common factor.

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

Step 6.5.5.2.3

Rewrite the expression.

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

Step 6.5.5.2.4

Divide $2$ by $1$ .

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Cancel the common factors.

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Step 7.2.1

Multiply by $1$ .

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

Step 7.2.2

Cancel the common factor.

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

Step 7.2.3

Rewrite the expression.

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

Step 7.2.4

Divide $y {\sqrt[3]{y^{2}}}^{2}$ by $1$ .

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

Step 8

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

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

Step 9

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

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Step 9.1

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

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

Step 9.2

Multiply $2$ by $2$ .

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

Step 10

Factor out $y^{3}$ .

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

Step 11

Pull terms out from under the radical.

$\log (y (y \sqrt[3]{y}))$

Step 12

Multiply $y$ by $y$ by adding the exponents.

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Step 12.1

Move $y$ .

$\log (y \cdot y \sqrt[3]{y})$

Step 12.2

Multiply $y$ by $y$ .

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