Precalculus Examples

$\frac{1}{3} \cdot \log_{b} (x^{4} y^{5}) - \frac{3}{4} \cdot \log_{b} (x^{2} y)$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{3} \log_{b} (x^{4} y^{5})$ by moving $\frac{1}{3}$ inside the logarithm.

$\log_{b} ({(x^{4} y^{5})}^{\frac{1}{3}}) - \frac{3}{4} \cdot \log_{b} (x^{2} y)$

Step 1.2

Apply the product rule to $x^{4} y^{5}$ .

$\log_{b} ({(x^{4})}^{\frac{1}{3}} {(y^{5})}^{\frac{1}{3}}) - \frac{3}{4} \cdot \log_{b} (x^{2} y)$

Step 1.3

Multiply the exponents in ${(x^{4})}^{\frac{1}{3}}$ .

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Step 1.3.1

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

$\log_{b} (x^{4 (\frac{1}{3})} {(y^{5})}^{\frac{1}{3}}) - \frac{3}{4} \cdot \log_{b} (x^{2} y)$

Step 1.3.2

Combine $4$ and $\frac{1}{3}$ .

$\log_{b} (x^{\frac{4}{3}} {(y^{5})}^{\frac{1}{3}}) - \frac{3}{4} \cdot \log_{b} (x^{2} y)$

Step 1.4

Multiply the exponents in ${(y^{5})}^{\frac{1}{3}}$ .

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Step 1.4.1

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

$\log_{b} (x^{\frac{4}{3}} y^{5 (\frac{1}{3})}) - \frac{3}{4} \cdot \log_{b} (x^{2} y)$

Step 1.4.2

Combine $5$ and $\frac{1}{3}$ .

$\log_{b} (x^{\frac{4}{3}} y^{\frac{5}{3}}) - \frac{3}{4} \cdot \log_{b} (x^{2} y)$

Step 1.5

Multiply $- \frac{3}{4} \log_{b} (x^{2} y)$ .

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Step 1.5.1

Reorder $\log_{b} (x^{2} y)$ and $\frac{3}{4}$ .

$\log_{b} (x^{\frac{4}{3}} y^{\frac{5}{3}}) - (\frac{3}{4} \log_{b} (x^{2} y))$

Step 1.5.2

Simplify $\frac{3}{4} \log_{b} (x^{2} y)$ by moving $\frac{3}{4}$ inside the logarithm.

$\log_{b} (x^{\frac{4}{3}} y^{\frac{5}{3}}) - \log_{b} ({(x^{2} y)}^{\frac{3}{4}})$

Step 1.6

Apply the product rule to $x^{2} y$ .

$\log_{b} (x^{\frac{4}{3}} y^{\frac{5}{3}}) - \log_{b} ({(x^{2})}^{\frac{3}{4}} y^{\frac{3}{4}})$

Step 1.7

Multiply the exponents in ${(x^{2})}^{\frac{3}{4}}$ .

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Step 1.7.1

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

$\log_{b} (x^{\frac{4}{3}} y^{\frac{5}{3}}) - \log_{b} (x^{2 (\frac{3}{4})} y^{\frac{3}{4}})$

Step 1.7.2

Cancel the common factor of $2$ .

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Step 1.7.2.1

Factor $2$ out of $4$ .

$\log_{b} (x^{\frac{4}{3}} y^{\frac{5}{3}}) - \log_{b} (x^{2 \frac{3}{2 (2)}} y^{\frac{3}{4}})$

Step 1.7.2.2

Cancel the common factor.

$\log_{b} (x^{\frac{4}{3}} y^{\frac{5}{3}}) - \log_{b} (x^{2 \frac{3}{2 \cdot 2}} y^{\frac{3}{4}})$

Step 1.7.2.3

Rewrite the expression.

$\log_{b} (x^{\frac{4}{3}} y^{\frac{5}{3}}) - \log_{b} (x^{\frac{3}{2}} y^{\frac{3}{4}})$

Step 2

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

$\log_{b} (\frac{x^{\frac{4}{3}} y^{\frac{5}{3}}}{x^{\frac{3}{2}} y^{\frac{3}{4}}})$

Step 3

Move $x^{\frac{4}{3}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\log_{b} (\frac{y^{\frac{5}{3}}}{x^{\frac{3}{2}} y^{\frac{3}{4}} x^{- \frac{4}{3}}})$

Step 4

Multiply $x^{\frac{3}{2}}$ by $x^{- \frac{4}{3}}$ by adding the exponents.

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Step 4.1

Move $x^{- \frac{4}{3}}$ .

$\log_{b} (\frac{y^{\frac{5}{3}}}{x^{- \frac{4}{3}} x^{\frac{3}{2}} y^{\frac{3}{4}}})$

Step 4.2

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

$\log_{b} (\frac{y^{\frac{5}{3}}}{x^{- \frac{4}{3} + \frac{3}{2}} y^{\frac{3}{4}}})$

Step 4.3

To write $- \frac{4}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\log_{b} (\frac{y^{\frac{5}{3}}}{x^{- \frac{4}{3} \cdot \frac{2}{2} + \frac{3}{2}} y^{\frac{3}{4}}})$

Step 4.4

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\log_{b} (\frac{y^{\frac{5}{3}}}{x^{- \frac{4}{3} \cdot \frac{2}{2} + \frac{3}{2} \cdot \frac{3}{3}} y^{\frac{3}{4}}})$

Step 4.5

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.5.1

Multiply $\frac{4}{3}$ by $\frac{2}{2}$ .

$\log_{b} (\frac{y^{\frac{5}{3}}}{x^{- \frac{4 \cdot 2}{3 \cdot 2} + \frac{3}{2} \cdot \frac{3}{3}} y^{\frac{3}{4}}})$

Step 4.5.2

Multiply $3$ by $2$ .

$\log_{b} (\frac{y^{\frac{5}{3}}}{x^{- \frac{4 \cdot 2}{6} + \frac{3}{2} \cdot \frac{3}{3}} y^{\frac{3}{4}}})$

Step 4.5.3

Multiply $\frac{3}{2}$ by $\frac{3}{3}$ .

$\log_{b} (\frac{y^{\frac{5}{3}}}{x^{- \frac{4 \cdot 2}{6} + \frac{3 \cdot 3}{2 \cdot 3}} y^{\frac{3}{4}}})$

Step 4.5.4

Multiply $2$ by $3$ .

$\log_{b} (\frac{y^{\frac{5}{3}}}{x^{- \frac{4 \cdot 2}{6} + \frac{3 \cdot 3}{6}} y^{\frac{3}{4}}})$

Step 4.6

Combine the numerators over the common denominator.

$\log_{b} (\frac{y^{\frac{5}{3}}}{x^{\frac{- 4 \cdot 2 + 3 \cdot 3}{6}} y^{\frac{3}{4}}})$

Step 4.7

Simplify the numerator.

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Step 4.7.1

Multiply $- 4$ by $2$ .

$\log_{b} (\frac{y^{\frac{5}{3}}}{x^{\frac{- 8 + 3 \cdot 3}{6}} y^{\frac{3}{4}}})$

Step 4.7.2

Multiply $3$ by $3$ .

$\log_{b} (\frac{y^{\frac{5}{3}}}{x^{\frac{- 8 + 9}{6}} y^{\frac{3}{4}}})$

Step 4.7.3

Add $- 8$ and $9$ .

$\log_{b} (\frac{y^{\frac{5}{3}}}{x^{\frac{1}{6}} y^{\frac{3}{4}}})$

Step 5

Move $y^{\frac{3}{4}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\log_{b} (\frac{y^{\frac{5}{3}} y^{- \frac{3}{4}}}{x^{\frac{1}{6}}})$

Step 6

Multiply $y^{\frac{5}{3}}$ by $y^{- \frac{3}{4}}$ by adding the exponents.

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Step 6.1

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

$\log_{b} (\frac{y^{\frac{5}{3} - \frac{3}{4}}}{x^{\frac{1}{6}}})$

Step 6.2

To write $\frac{5}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\log_{b} (\frac{y^{\frac{5}{3} \cdot \frac{4}{4} - \frac{3}{4}}}{x^{\frac{1}{6}}})$

Step 6.3

To write $- \frac{3}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\log_{b} (\frac{y^{\frac{5}{3} \cdot \frac{4}{4} - \frac{3}{4} \cdot \frac{3}{3}}}{x^{\frac{1}{6}}})$

Step 6.4

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.4.1

Multiply $\frac{5}{3}$ by $\frac{4}{4}$ .

$\log_{b} (\frac{y^{\frac{5 \cdot 4}{3 \cdot 4} - \frac{3}{4} \cdot \frac{3}{3}}}{x^{\frac{1}{6}}})$

Step 6.4.2

Multiply $3$ by $4$ .

$\log_{b} (\frac{y^{\frac{5 \cdot 4}{12} - \frac{3}{4} \cdot \frac{3}{3}}}{x^{\frac{1}{6}}})$

Step 6.4.3

Multiply $\frac{3}{4}$ by $\frac{3}{3}$ .

$\log_{b} (\frac{y^{\frac{5 \cdot 4}{12} - \frac{3 \cdot 3}{4 \cdot 3}}}{x^{\frac{1}{6}}})$

Step 6.4.4

Multiply $4$ by $3$ .

$\log_{b} (\frac{y^{\frac{5 \cdot 4}{12} - \frac{3 \cdot 3}{12}}}{x^{\frac{1}{6}}})$

Step 6.5

Combine the numerators over the common denominator.

$\log_{b} (\frac{y^{\frac{5 \cdot 4 - 3 \cdot 3}{12}}}{x^{\frac{1}{6}}})$

Step 6.6

Simplify the numerator.

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Step 6.6.1

Multiply $5$ by $4$ .

$\log_{b} (\frac{y^{\frac{20 - 3 \cdot 3}{12}}}{x^{\frac{1}{6}}})$

Step 6.6.2

Multiply $- 3$ by $3$ .

$\log_{b} (\frac{y^{\frac{20 - 9}{12}}}{x^{\frac{1}{6}}})$

Step 6.6.3

Subtract $9$ from $20$ .

$\log_{b} (\frac{y^{\frac{11}{12}}}{x^{\frac{1}{6}}})$