Precalculus Examples

$\log (\sqrt[5]{\frac{a - 6}{4 a}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{\frac{a - 6}{4 a}}$ as ${(\frac{a - 6}{4 a})}^{\frac{1}{5}}$ .

$\log ({(\frac{a - 6}{4 a})}^{\frac{1}{5}})$

Step 2

Expand $\log ({(\frac{a - 6}{4 a})}^{\frac{1}{5}})$ by moving $\frac{1}{5}$ outside the logarithm.

$\frac{1}{5} \log (\frac{a - 6}{4 a})$

Step 3

Rewrite $\log (\frac{a - 6}{4 a})$ as $\log (a - 6) - \log (4 a)$ .

$\frac{1}{5} (\log (a - 6) - \log (4 a))$

Step 4

Rewrite $\log (4 a)$ as $\log (4) + \log (a)$ .

$\frac{1}{5} (\log (a - 6) - (\log (4) + \log (a)))$

Step 5

Simplify each term.

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Step 5.1

Simplify each term.

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Step 5.1.1

Rewrite $\log (4)$ as $\log (2^{2})$ .

$\frac{1}{5} (\log (a - 6) - (\log (2^{2}) + \log (a)))$

Step 5.1.2

Expand $\log (2^{2})$ by moving $2$ outside the logarithm.

$\frac{1}{5} (\log (a - 6) - (2 \log (2) + \log (a)))$

Step 5.2

Apply the distributive property.

$\frac{1}{5} (\log (a - 6) - (2 \log (2)) - \log (a))$

Step 5.3

Multiply $2$ by $- 1$ .

$\frac{1}{5} (\log (a - 6) - 2 \log (2) - \log (a))$

Step 6

Apply the distributive property.

$\frac{1}{5} \log (a - 6) + \frac{1}{5} (- 2 \log (2)) + \frac{1}{5} (- \log (a))$

Step 7

Simplify.

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Step 7.1

Combine $\frac{1}{5}$ and $\log (a - 6)$ .

$\frac{\log (a - 6)}{5} + \frac{1}{5} (- 2 \log (2)) + \frac{1}{5} (- \log (a))$

Step 7.2

Multiply $\frac{1}{5} (- 2 \log (2))$ .

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Step 7.2.1

Combine $- 2$ and $\frac{1}{5}$ .

$\frac{\log (a - 6)}{5} + \frac{- 2}{5} \log (2) + \frac{1}{5} (- \log (a))$

Step 7.2.2

Combine $\frac{- 2}{5}$ and $\log (2)$ .

$\frac{\log (a - 6)}{5} + \frac{- 2 \log (2)}{5} + \frac{1}{5} (- \log (a))$

Step 7.3

Combine $\frac{1}{5}$ and $\log (a)$ .

$\frac{\log (a - 6)}{5} + \frac{- 2 \log (2)}{5} - \frac{\log (a)}{5}$

Step 8

Move the negative in front of the fraction.

$\frac{\log (a - 6)}{5} - \frac{2 \log (2)}{5} - \frac{\log (a)}{5}$

Step 9

Combine the numerators over the common denominator.

$\frac{\log (a - 6) - 2 \log (2)}{5} - \frac{\log (a)}{5}$