Precalculus Examples

$\log (\sqrt[4]{\frac{m^{8} n^{8}}{c^{9}}})$

Step 1

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

$\log ({(\frac{m^{8} n^{8}}{c^{9}})}^{\frac{1}{4}})$

Step 2

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

$\frac{1}{4} \log (\frac{m^{8} n^{8}}{c^{9}})$

Step 3

Rewrite $\log (\frac{m^{8} n^{8}}{c^{9}})$ as $\log (m^{8} n^{8}) - \log (c^{9})$ .

$\frac{1}{4} (\log (m^{8} n^{8}) - \log (c^{9}))$

Step 4

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

$\frac{1}{4} (\log (m^{8} n^{8}) - (9 \log (c)))$

Step 5

Multiply $9$ by $- 1$ .

$\frac{1}{4} (\log (m^{8} n^{8}) - 9 \log (c))$

Step 6

Rewrite $\log (m^{8} n^{8})$ as $\log (m^{8}) + \log (n^{8})$ .

$\frac{1}{4} (\log (m^{8}) + \log (n^{8}) - 9 \log (c))$

Step 7

Expand $\log (m^{8})$ by moving $8$ outside the logarithm.

$\frac{1}{4} (8 \log (m) + \log (n^{8}) - 9 \log (c))$

Step 8

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

$\frac{1}{4} (8 \log (m) + 8 \log (n) - 9 \log (c))$

Step 9

Apply the distributive property.

$\frac{1}{4} (8 \log (m)) + \frac{1}{4} (8 \log (n)) + \frac{1}{4} (- 9 \log (c))$

Step 10

Simplify.

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Step 10.1

Cancel the common factor of $4$ .

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Step 10.1.1

Factor $4$ out of $8 \log (m)$ .

$\frac{1}{4} (4 (2 \log (m))) + \frac{1}{4} (8 \log (n)) + \frac{1}{4} (- 9 \log (c))$

Step 10.1.2

Cancel the common factor.

$\frac{1}{4} (4 (2 \log (m))) + \frac{1}{4} (8 \log (n)) + \frac{1}{4} (- 9 \log (c))$

Step 10.1.3

Rewrite the expression.

$2 \log (m) + \frac{1}{4} (8 \log (n)) + \frac{1}{4} (- 9 \log (c))$

Step 10.2

Cancel the common factor of $4$ .

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Step 10.2.1

Factor $4$ out of $8 \log (n)$ .

$2 \log (m) + \frac{1}{4} (4 (2 \log (n))) + \frac{1}{4} (- 9 \log (c))$

Step 10.2.2

Cancel the common factor.

$2 \log (m) + \frac{1}{4} (4 (2 \log (n))) + \frac{1}{4} (- 9 \log (c))$

Step 10.2.3

Rewrite the expression.

$2 \log (m) + 2 \log (n) + \frac{1}{4} (- 9 \log (c))$

Step 10.3

Multiply $\frac{1}{4} (- 9 \log (c))$ .

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Step 10.3.1

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

$2 \log (m) + 2 \log (n) + \frac{- 9}{4} \log (c)$

Step 10.3.2

Combine $\frac{- 9}{4}$ and $\log (c)$ .

$2 \log (m) + 2 \log (n) + \frac{- 9 \log (c)}{4}$

Step 11

Move the negative in front of the fraction.

$2 \log (m) + 2 \log (n) - \frac{9 \log (c)}{4}$