Precalculus Examples

$\frac{1}{2} \cdot \log_{y} (p^{3} q^{4}) - \frac{2}{3} \cdot \log_{y} (p^{4} q^{3})$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{2} \log_{y} (p^{3} q^{4})$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{y} ({(p^{3} q^{4})}^{\frac{1}{2}}) - \frac{2}{3} \cdot \log_{y} (p^{4} q^{3})$

Step 1.2

Apply the product rule to $p^{3} q^{4}$ .

$\log_{y} ({(p^{3})}^{\frac{1}{2}} {(q^{4})}^{\frac{1}{2}}) - \frac{2}{3} \cdot \log_{y} (p^{4} q^{3})$

Step 1.3

Multiply the exponents in ${(p^{3})}^{\frac{1}{2}}$ .

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Step 1.3.1

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

$\log_{y} (p^{3 (\frac{1}{2})} {(q^{4})}^{\frac{1}{2}}) - \frac{2}{3} \cdot \log_{y} (p^{4} q^{3})$

Step 1.3.2

Combine $3$ and $\frac{1}{2}$ .

$\log_{y} (p^{\frac{3}{2}} {(q^{4})}^{\frac{1}{2}}) - \frac{2}{3} \cdot \log_{y} (p^{4} q^{3})$

Step 1.4

Multiply the exponents in ${(q^{4})}^{\frac{1}{2}}$ .

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Step 1.4.1

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

$\log_{y} (p^{\frac{3}{2}} q^{4 (\frac{1}{2})}) - \frac{2}{3} \cdot \log_{y} (p^{4} q^{3})$

Step 1.4.2

Cancel the common factor of $2$ .

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Step 1.4.2.1

Factor $2$ out of $4$ .

$\log_{y} (p^{\frac{3}{2}} q^{2 (2) \frac{1}{2}}) - \frac{2}{3} \cdot \log_{y} (p^{4} q^{3})$

Step 1.4.2.2

Cancel the common factor.

$\log_{y} (p^{\frac{3}{2}} q^{2 \cdot 2 \frac{1}{2}}) - \frac{2}{3} \cdot \log_{y} (p^{4} q^{3})$

Step 1.4.2.3

Rewrite the expression.

$\log_{y} (p^{\frac{3}{2}} q^{2}) - \frac{2}{3} \cdot \log_{y} (p^{4} q^{3})$

Step 1.5

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

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Step 1.5.1

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

$\log_{y} (p^{\frac{3}{2}} q^{2}) - (\frac{2}{3} \log_{y} (p^{4} q^{3}))$

Step 1.5.2

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

$\log_{y} (p^{\frac{3}{2}} q^{2}) - \log_{y} ({(p^{4} q^{3})}^{\frac{2}{3}})$

Step 1.6

Apply the product rule to $p^{4} q^{3}$ .

$\log_{y} (p^{\frac{3}{2}} q^{2}) - \log_{y} ({(p^{4})}^{\frac{2}{3}} {(q^{3})}^{\frac{2}{3}})$

Step 1.7

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

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Step 1.7.1

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

$\log_{y} (p^{\frac{3}{2}} q^{2}) - \log_{y} (p^{4 (\frac{2}{3})} {(q^{3})}^{\frac{2}{3}})$

Step 1.7.2

Multiply $4 (\frac{2}{3})$ .

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Step 1.7.2.1

Combine $4$ and $\frac{2}{3}$ .

$\log_{y} (p^{\frac{3}{2}} q^{2}) - \log_{y} (p^{\frac{4 \cdot 2}{3}} {(q^{3})}^{\frac{2}{3}})$

Step 1.7.2.2

Multiply $4$ by $2$ .

$\log_{y} (p^{\frac{3}{2}} q^{2}) - \log_{y} (p^{\frac{8}{3}} {(q^{3})}^{\frac{2}{3}})$

Step 1.8

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

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Step 1.8.1

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

$\log_{y} (p^{\frac{3}{2}} q^{2}) - \log_{y} (p^{\frac{8}{3}} q^{3 (\frac{2}{3})})$

Step 1.8.2

Cancel the common factor of $3$ .

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Step 1.8.2.1

Cancel the common factor.

$\log_{y} (p^{\frac{3}{2}} q^{2}) - \log_{y} (p^{\frac{8}{3}} q^{3 (\frac{2}{3})})$

Step 1.8.2.2

Rewrite the expression.

$\log_{y} (p^{\frac{3}{2}} q^{2}) - \log_{y} (p^{\frac{8}{3}} q^{2})$

Step 2

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

$\log_{y} (\frac{p^{\frac{3}{2}} q^{2}}{p^{\frac{8}{3}} q^{2}})$

Step 3

Move $p^{\frac{3}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\log_{y} (\frac{q^{2}}{p^{\frac{8}{3}} q^{2} p^{- \frac{3}{2}}})$

Step 4

Multiply $p^{\frac{8}{3}}$ by $p^{- \frac{3}{2}}$ by adding the exponents.

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Step 4.1

Move $p^{- \frac{3}{2}}$ .

$\log_{y} (\frac{q^{2}}{p^{- \frac{3}{2}} p^{\frac{8}{3}} q^{2}})$

Step 4.2

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

$\log_{y} (\frac{q^{2}}{p^{- \frac{3}{2} + \frac{8}{3}} q^{2}})$

Step 4.3

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

$\log_{y} (\frac{q^{2}}{p^{- \frac{3}{2} \cdot \frac{3}{3} + \frac{8}{3}} q^{2}})$

Step 4.4

To write $\frac{8}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\log_{y} (\frac{q^{2}}{p^{- \frac{3}{2} \cdot \frac{3}{3} + \frac{8}{3} \cdot \frac{2}{2}} q^{2}})$

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

$\log_{y} (\frac{q^{2}}{p^{- \frac{3 \cdot 3}{2 \cdot 3} + \frac{8}{3} \cdot \frac{2}{2}} q^{2}})$

Step 4.5.2

Multiply $2$ by $3$ .

$\log_{y} (\frac{q^{2}}{p^{- \frac{3 \cdot 3}{6} + \frac{8}{3} \cdot \frac{2}{2}} q^{2}})$

Step 4.5.3

Multiply $\frac{8}{3}$ by $\frac{2}{2}$ .

$\log_{y} (\frac{q^{2}}{p^{- \frac{3 \cdot 3}{6} + \frac{8 \cdot 2}{3 \cdot 2}} q^{2}})$

Step 4.5.4

Multiply $3$ by $2$ .

$\log_{y} (\frac{q^{2}}{p^{- \frac{3 \cdot 3}{6} + \frac{8 \cdot 2}{6}} q^{2}})$

Step 4.6

Combine the numerators over the common denominator.

$\log_{y} (\frac{q^{2}}{p^{\frac{- 3 \cdot 3 + 8 \cdot 2}{6}} q^{2}})$

Step 4.7

Simplify the numerator.

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Step 4.7.1

Multiply $- 3$ by $3$ .

$\log_{y} (\frac{q^{2}}{p^{\frac{- 9 + 8 \cdot 2}{6}} q^{2}})$

Step 4.7.2

Multiply $8$ by $2$ .

$\log_{y} (\frac{q^{2}}{p^{\frac{- 9 + 16}{6}} q^{2}})$

Step 4.7.3

Add $- 9$ and $16$ .

$\log_{y} (\frac{q^{2}}{p^{\frac{7}{6}} q^{2}})$

Step 5

Cancel the common factor.

$\log_{y} (\frac{q^{2}}{p^{\frac{7}{6}} q^{2}})$

Step 6

Rewrite the expression.

$\log_{y} (\frac{1}{p^{\frac{7}{6}}})$