Precalculus Examples

$\log_{c} (\sqrt[4]{\frac{m^{16} n^{20}}{c^{3}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{\frac{m^{16} n^{20}}{c^{3}}}$ as ${(\frac{m^{16} n^{20}}{c^{3}})}^{\frac{1}{4}}$ .

$\log_{c} ({(\frac{m^{16} n^{20}}{c^{3}})}^{\frac{1}{4}})$

Step 2

Expand $\log_{c} ({(\frac{m^{16} n^{20}}{c^{3}})}^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

$\frac{1}{4} \log_{c} (\frac{m^{16} n^{20}}{c^{3}})$

Step 3

Rewrite $\log_{c} (\frac{m^{16} n^{20}}{c^{3}})$ as $\log_{c} (m^{16} n^{20}) - \log_{c} (c^{3})$ .

$\frac{1}{4} (\log_{c} (m^{16} n^{20}) - \log_{c} (c^{3}))$

Step 4

Expand $\log_{c} (c^{3})$ by moving $3$ outside the logarithm.

$\frac{1}{4} (\log_{c} (m^{16} n^{20}) - (3 \log_{c} (c)))$

Step 5

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

$\frac{1}{4} (\log_{c} (m^{16} n^{20}) - (3 \cdot 1))$

Step 6

Multiply $3$ by $1$ .

$\frac{1}{4} (\log_{c} (m^{16} n^{20}) - 1 \cdot 3)$

Step 7

Multiply $- 1$ by $3$ .

$\frac{1}{4} (\log_{c} (m^{16} n^{20}) - 3)$

Step 8

Rewrite $\log_{c} (m^{16} n^{20})$ as $\log_{c} (m^{16}) + \log_{c} (n^{20})$ .

$\frac{1}{4} (\log_{c} (m^{16}) + \log_{c} (n^{20}) - 3)$

Step 9

Expand $\log_{c} (m^{16})$ by moving $16$ outside the logarithm.

$\frac{1}{4} (16 \log_{c} (m) + \log_{c} (n^{20}) - 3)$

Step 10

Expand $\log_{c} (n^{20})$ by moving $20$ outside the logarithm.

$\frac{1}{4} (16 \log_{c} (m) + 20 \log_{c} (n) - 3)$

Step 11

Apply the distributive property.

$\frac{1}{4} (16 \log_{c} (m)) + \frac{1}{4} (20 \log_{c} (n)) + \frac{1}{4} \cdot - 3$

Step 12

Simplify.

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Step 12.1

Cancel the common factor of $4$ .

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Step 12.1.1

Factor $4$ out of $16 \log_{c} (m)$ .

$\frac{1}{4} (4 (4 \log_{c} (m))) + \frac{1}{4} (20 \log_{c} (n)) + \frac{1}{4} \cdot - 3$

Step 12.1.2

Cancel the common factor.

$\frac{1}{4} (4 (4 \log_{c} (m))) + \frac{1}{4} (20 \log_{c} (n)) + \frac{1}{4} \cdot - 3$

Step 12.1.3

Rewrite the expression.

$4 \log_{c} (m) + \frac{1}{4} (20 \log_{c} (n)) + \frac{1}{4} \cdot - 3$

Step 12.2

Cancel the common factor of $4$ .

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Step 12.2.1

Factor $4$ out of $20 \log_{c} (n)$ .

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

Step 12.2.2

Cancel the common factor.

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

Step 12.2.3

Rewrite the expression.

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

Step 12.3

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

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

Step 13

Move the negative in front of the fraction.

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