Precalculus Examples

$\log_{c} (\sqrt[4]{\frac{m^{12} n^{8}}{c^{9} a^{5}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{\frac{m^{12} n^{8}}{c^{9} a^{5}}}$ as ${(\frac{m^{12} n^{8}}{c^{9} a^{5}})}^{\frac{1}{4}}$ .

$\log_{c} ({(\frac{m^{12} n^{8}}{c^{9} a^{5}})}^{\frac{1}{4}})$

Step 2

Expand $\log_{c} ({(\frac{m^{12} n^{8}}{c^{9} a^{5}})}^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

$\frac{1}{4} \log_{c} (\frac{m^{12} n^{8}}{c^{9} a^{5}})$

Step 3

Rewrite $\log_{c} (\frac{m^{12} n^{8}}{c^{9} a^{5}})$ as $\log_{c} (m^{12} n^{8}) - \log_{c} (c^{9} a^{5})$ .

$\frac{1}{4} (\log_{c} (m^{12} n^{8}) - \log_{c} (c^{9} a^{5}))$

Step 4

Rewrite $\log_{c} (m^{12} n^{8})$ as $\log_{c} (m^{12}) + \log_{c} (n^{8})$ .

$\frac{1}{4} (\log_{c} (m^{12}) + \log_{c} (n^{8}) - \log_{c} (c^{9} a^{5}))$

Step 5

Expand $\log_{c} (m^{12})$ by moving $12$ outside the logarithm.

$\frac{1}{4} (12 \log_{c} (m) + \log_{c} (n^{8}) - \log_{c} (c^{9} a^{5}))$

Step 6

Expand $\log_{c} (n^{8})$ by moving $8$ outside the logarithm.

$\frac{1}{4} (12 \log_{c} (m) + 8 \log_{c} (n) - \log_{c} (c^{9} a^{5}))$

Step 7

Rewrite $\log_{c} (c^{9} a^{5})$ as $\log_{c} (c^{9}) + \log_{c} (a^{5})$ .

$\frac{1}{4} (12 \log_{c} (m) + 8 \log_{c} (n) - (\log_{c} (c^{9}) + \log_{c} (a^{5})))$

Step 8

Expand $\log_{c} (c^{9})$ by moving $9$ outside the logarithm.

$\frac{1}{4} (12 \log_{c} (m) + 8 \log_{c} (n) - (9 \log_{c} (c) + \log_{c} (a^{5})))$

Step 9

Expand $\log_{c} (a^{5})$ by moving $5$ outside the logarithm.

$\frac{1}{4} (12 \log_{c} (m) + 8 \log_{c} (n) - (9 \log_{c} (c) + 5 \log_{c} (a)))$

Step 10

Logarithm base $c$ of $c$ is $1$ .

$\frac{1}{4} (12 \log_{c} (m) + 8 \log_{c} (n) - (9 \cdot 1 + 5 \log_{c} (a)))$

Step 11

Multiply $9$ by $1$ .

$\frac{1}{4} (12 \log_{c} (m) + 8 \log_{c} (n) - (9 + 5 \log_{c} (a)))$

Step 12

Simplify each term.

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Step 12.1

Apply the distributive property.

$\frac{1}{4} (12 \log_{c} (m) + 8 \log_{c} (n) - 1 \cdot 9 - (5 \log_{c} (a)))$

Step 12.2

Multiply $- 1$ by $9$ .

$\frac{1}{4} (12 \log_{c} (m) + 8 \log_{c} (n) - 9 - (5 \log_{c} (a)))$

Step 12.3

Multiply $5$ by $- 1$ .

$\frac{1}{4} (12 \log_{c} (m) + 8 \log_{c} (n) - 9 - 5 \log_{c} (a))$

Step 13

Apply the distributive property.

$\frac{1}{4} (12 \log_{c} (m)) + \frac{1}{4} (8 \log_{c} (n)) + \frac{1}{4} \cdot - 9 + \frac{1}{4} (- 5 \log_{c} (a))$

Step 14

Simplify.

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Step 14.1

Cancel the common factor of $4$ .

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Step 14.1.1

Factor $4$ out of $12 \log_{c} (m)$ .

$\frac{1}{4} (4 (3 \log_{c} (m))) + \frac{1}{4} (8 \log_{c} (n)) + \frac{1}{4} \cdot - 9 + \frac{1}{4} (- 5 \log_{c} (a))$

Step 14.1.2

Cancel the common factor.

$\frac{1}{4} (4 (3 \log_{c} (m))) + \frac{1}{4} (8 \log_{c} (n)) + \frac{1}{4} \cdot - 9 + \frac{1}{4} (- 5 \log_{c} (a))$

Step 14.1.3

Rewrite the expression.

$3 \log_{c} (m) + \frac{1}{4} (8 \log_{c} (n)) + \frac{1}{4} \cdot - 9 + \frac{1}{4} (- 5 \log_{c} (a))$

Step 14.2

Cancel the common factor of $4$ .

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Step 14.2.1

Factor $4$ out of $8 \log_{c} (n)$ .

$3 \log_{c} (m) + \frac{1}{4} (4 (2 \log_{c} (n))) + \frac{1}{4} \cdot - 9 + \frac{1}{4} (- 5 \log_{c} (a))$

Step 14.2.2

Cancel the common factor.

$3 \log_{c} (m) + \frac{1}{4} (4 (2 \log_{c} (n))) + \frac{1}{4} \cdot - 9 + \frac{1}{4} (- 5 \log_{c} (a))$

Step 14.2.3

Rewrite the expression.

$3 \log_{c} (m) + 2 \log_{c} (n) + \frac{1}{4} \cdot - 9 + \frac{1}{4} (- 5 \log_{c} (a))$

Step 14.3

Combine $\frac{1}{4}$ and $- 9$ .

$3 \log_{c} (m) + 2 \log_{c} (n) + \frac{- 9}{4} + \frac{1}{4} (- 5 \log_{c} (a))$

Step 14.4

Multiply $\frac{1}{4} (- 5 \log_{c} (a))$ .

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Step 14.4.1

Combine $- 5$ and $\frac{1}{4}$ .

$3 \log_{c} (m) + 2 \log_{c} (n) + \frac{- 9}{4} + \frac{- 5}{4} \log_{c} (a)$

Step 14.4.2

Combine $\frac{- 5}{4}$ and $\log_{c} (a)$ .

$3 \log_{c} (m) + 2 \log_{c} (n) + \frac{- 9}{4} + \frac{- 5 \log_{c} (a)}{4}$

Step 15

Simplify each term.

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Step 15.1

Move the negative in front of the fraction.

$3 \log_{c} (m) + 2 \log_{c} (n) - \frac{9}{4} + \frac{- 5 \log_{c} (a)}{4}$

Step 15.2

Move the negative in front of the fraction.

$3 \log_{c} (m) + 2 \log_{c} (n) - \frac{9}{4} - \frac{5 \log_{c} (a)}{4}$