Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify $- 2 \log (x)$ by moving $2$ inside the logarithm.

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

Simplify each term.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

Multiply $y^{2}$ by $y^{3}$ by adding the exponents.

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Step 3.3.1

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

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

Step 3.3.2

Add $2$ and $3$ .

$- \log (x^{2}) \sqrt[3]{y} + \log (y^{5})$