Precalculus Examples

$\frac{1}{5} \cdot \log_{3} (\frac{a^{6} b}{9})$

Step 1

Rewrite $\log_{3} (\frac{a^{6} b}{9})$ as $\log_{3} (a^{6} b) - \log_{3} (9)$ .

$\frac{1}{5} \cdot (\log_{3} (a^{6} b) - \log_{3} (9))$

Step 2

Logarithm base $3$ of $9$ is $2$ .

$\frac{1}{5} \cdot (\log_{3} (a^{6} b) - 1 \cdot 2)$

Step 3

Multiply $- 1$ by $2$ .

$\frac{1}{5} \cdot (\log_{3} (a^{6} b) - 2)$

Step 4

Rewrite $\log_{3} (a^{6} b)$ as $\log_{3} (a^{6}) + \log_{3} (b)$ .

$\frac{1}{5} \cdot (\log_{3} (a^{6}) + \log_{3} (b) - 2)$

Step 5

Expand $\log_{3} (a^{6})$ by moving $6$ outside the logarithm.

$\frac{1}{5} \cdot (6 \log_{3} (a) + \log_{3} (b) - 2)$

Step 6

Apply the distributive property.

$\frac{1}{5} (6 \log_{3} (a)) + \frac{1}{5} \log_{3} (b) + \frac{1}{5} \cdot - 2$

Step 7

Simplify.

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Step 7.1

Multiply $\frac{1}{5} (6 \log_{3} (a))$ .

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Step 7.1.1

Combine $6$ and $\frac{1}{5}$ .

$\frac{6}{5} \log_{3} (a) + \frac{1}{5} \log_{3} (b) + \frac{1}{5} \cdot - 2$

Step 7.1.2

Combine $\frac{6}{5}$ and $\log_{3} (a)$ .

$\frac{6 \log_{3} (a)}{5} + \frac{1}{5} \log_{3} (b) + \frac{1}{5} \cdot - 2$

Step 7.2

Combine $\frac{1}{5}$ and $\log_{3} (b)$ .

$\frac{6 \log_{3} (a)}{5} + \frac{\log_{3} (b)}{5} + \frac{1}{5} \cdot - 2$

Step 7.3

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

$\frac{6 \log_{3} (a)}{5} + \frac{\log_{3} (b)}{5} + \frac{- 2}{5}$

Step 8

Move the negative in front of the fraction.

$\frac{6 \log_{3} (a)}{5} + \frac{\log_{3} (b)}{5} - \frac{2}{5}$