Precalculus Examples

$\frac{1}{4} \cdot \log (c x) + 5 \log (c y) - 4 \log (c x)$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{4} \log (c x)$ by moving $\frac{1}{4}$ inside the logarithm.

$\log ({(c x)}^{\frac{1}{4}}) + 5 \log (c y) - 4 \log (c x)$

Step 1.2

Apply the product rule to $c x$ .

$\log (c^{\frac{1}{4}} x^{\frac{1}{4}}) + 5 \log (c y) - 4 \log (c x)$

Step 1.3

Simplify $5 \log (c y)$ by moving $5$ inside the logarithm.

$\log (c^{\frac{1}{4}} x^{\frac{1}{4}}) + \log ({(c y)}^{5}) - 4 \log (c x)$

Step 1.4

Apply the product rule to $c y$ .

$\log (c^{\frac{1}{4}} x^{\frac{1}{4}}) + \log (c^{5} y^{5}) - 4 \log (c x)$

Step 1.5

Simplify $- 4 \log (c x)$ by moving $4$ inside the logarithm.

$\log (c^{\frac{1}{4}} x^{\frac{1}{4}}) + \log (c^{5} y^{5}) - \log ({(c x)}^{4})$

Step 1.6

Apply the product rule to $c x$ .

$\log (c^{\frac{1}{4}} x^{\frac{1}{4}}) + \log (c^{5} y^{5}) - \log (c^{4} x^{4})$

Step 2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log (c^{\frac{1}{4}} x^{\frac{1}{4}} (c^{5} y^{5})) - \log (c^{4} x^{4})$

Step 3

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

$\log (\frac{c^{\frac{1}{4}} x^{\frac{1}{4}} (c^{5} y^{5})}{c^{4} x^{4}})$

Step 4

Multiply $c^{\frac{1}{4}}$ by $c^{5}$ by adding the exponents.

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Step 4.1

Move $c^{5}$ .

$\log (\frac{c^{5} c^{\frac{1}{4}} x^{\frac{1}{4}} y^{5}}{c^{4} x^{4}})$

Step 4.2

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

$\log (\frac{c^{5 + \frac{1}{4}} x^{\frac{1}{4}} y^{5}}{c^{4} x^{4}})$

Step 4.3

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

$\log (\frac{c^{5 \cdot \frac{4}{4} + \frac{1}{4}} x^{\frac{1}{4}} y^{5}}{c^{4} x^{4}})$

Step 4.4

Combine $5$ and $\frac{4}{4}$ .

$\log (\frac{c^{\frac{5 \cdot 4}{4} + \frac{1}{4}} x^{\frac{1}{4}} y^{5}}{c^{4} x^{4}})$

Step 4.5

Combine the numerators over the common denominator.

$\log (\frac{c^{\frac{5 \cdot 4 + 1}{4}} x^{\frac{1}{4}} y^{5}}{c^{4} x^{4}})$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $5$ by $4$ .

$\log (\frac{c^{\frac{20 + 1}{4}} x^{\frac{1}{4}} y^{5}}{c^{4} x^{4}})$

Step 4.6.2

Add $20$ and $1$ .

$\log (\frac{c^{\frac{21}{4}} x^{\frac{1}{4}} y^{5}}{c^{4} x^{4}})$

Step 5

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

$\log (\frac{c^{\frac{21}{4}} y^{5}}{c^{4} x^{4} x^{- \frac{1}{4}}})$

Step 6

Multiply $x^{4}$ by $x^{- \frac{1}{4}}$ by adding the exponents.

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Step 6.1

Move $x^{- \frac{1}{4}}$ .

$\log (\frac{c^{\frac{21}{4}} y^{5}}{c^{4} (x^{- \frac{1}{4}} x^{4})})$

Step 6.2

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

$\log (\frac{c^{\frac{21}{4}} y^{5}}{c^{4} x^{- \frac{1}{4} + 4}})$

Step 6.3

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

$\log (\frac{c^{\frac{21}{4}} y^{5}}{c^{4} x^{- \frac{1}{4} + 4 \cdot \frac{4}{4}}})$

Step 6.4

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

$\log (\frac{c^{\frac{21}{4}} y^{5}}{c^{4} x^{- \frac{1}{4} + \frac{4 \cdot 4}{4}}})$

Step 6.5

Combine the numerators over the common denominator.

$\log (\frac{c^{\frac{21}{4}} y^{5}}{c^{4} x^{\frac{- 1 + 4 \cdot 4}{4}}})$

Step 6.6

Simplify the numerator.

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Step 6.6.1

Multiply $4$ by $4$ .

$\log (\frac{c^{\frac{21}{4}} y^{5}}{c^{4} x^{\frac{- 1 + 16}{4}}})$

Step 6.6.2

Add $- 1$ and $16$ .

$\log (\frac{c^{\frac{21}{4}} y^{5}}{c^{4} x^{\frac{15}{4}}})$

Step 7

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

$\log (\frac{c^{\frac{21}{4}} y^{5} c^{- 4}}{x^{\frac{15}{4}}})$

Step 8

Multiply $c^{\frac{21}{4}}$ by $c^{- 4}$ by adding the exponents.

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Step 8.1

Move $c^{- 4}$ .

$\log (\frac{c^{- 4} c^{\frac{21}{4}} y^{5}}{x^{\frac{15}{4}}})$

Step 8.2

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

$\log (\frac{c^{- 4 + \frac{21}{4}} y^{5}}{x^{\frac{15}{4}}})$

Step 8.3

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

$\log (\frac{c^{- 4 \cdot \frac{4}{4} + \frac{21}{4}} y^{5}}{x^{\frac{15}{4}}})$

Step 8.4

Combine $- 4$ and $\frac{4}{4}$ .

$\log (\frac{c^{\frac{- 4 \cdot 4}{4} + \frac{21}{4}} y^{5}}{x^{\frac{15}{4}}})$

Step 8.5

Combine the numerators over the common denominator.

$\log (\frac{c^{\frac{- 4 \cdot 4 + 21}{4}} y^{5}}{x^{\frac{15}{4}}})$

Step 8.6

Simplify the numerator.

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Step 8.6.1

Multiply $- 4$ by $4$ .

$\log (\frac{c^{\frac{- 16 + 21}{4}} y^{5}}{x^{\frac{15}{4}}})$

Step 8.6.2

Add $- 16$ and $21$ .

$\log (\frac{c^{\frac{5}{4}} y^{5}}{x^{\frac{15}{4}}})$

Step 9

Reorder factors in $\log (\frac{c^{\frac{5}{4}} y^{5}}{x^{\frac{15}{4}}})$ .

$\log (\frac{y^{5} c^{\frac{5}{4}}}{x^{\frac{15}{4}}})$