Precalculus Examples

$\log_{3} (\sqrt{\frac{r q}{6}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{r q}{6}}$ as ${(\frac{r q}{6})}^{\frac{1}{2}}$ .

$\log_{3} ({(\frac{r q}{6})}^{\frac{1}{2}})$

Step 2

Expand $\log_{3} ({(\frac{r q}{6})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{3} (\frac{r q}{6})$

Step 3

Rewrite $\log_{3} (\frac{r q}{6})$ as $\log_{3} (r q) - \log_{3} (6)$ .

$\frac{1}{2} (\log_{3} (r q) - \log_{3} (6))$

Step 4

Rewrite $\log_{3} (6)$ as $\log_{3} (2 (3))$ .

$\frac{1}{2} (\log_{3} (r q) - \log_{3} (2 (3)))$

Step 5

Rewrite $\log_{3} (2 (3))$ as $\log_{3} (2) + \log_{3} (3)$ .

$\frac{1}{2} (\log_{3} (r q) - (\log_{3} (2) + \log_{3} (3)))$

Step 6

Logarithm base $3$ of $3$ is $1$ .

$\frac{1}{2} (\log_{3} (r q) - (\log_{3} (2) + 1))$

Step 7

Apply the distributive property.

$\frac{1}{2} (\log_{3} (r q) - \log_{3} (2) - 1 \cdot 1)$

Step 8

Multiply $- 1$ by $1$ .

$\frac{1}{2} (\log_{3} (r q) - \log_{3} (2) - 1)$

Step 9

Rewrite $\log_{3} (r q)$ as $\log_{3} (r) + \log_{3} (q)$ .

$\frac{1}{2} (\log_{3} (r) + \log_{3} (q) - \log_{3} (2) - 1)$

Step 10

Apply the distributive property.

$\frac{1}{2} \log_{3} (r) + \frac{1}{2} \log_{3} (q) + \frac{1}{2} (- \log_{3} (2)) + \frac{1}{2} \cdot - 1$

Step 11

Simplify.

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Step 11.1

Combine $\frac{1}{2}$ and $\log_{3} (r)$ .

$\frac{\log_{3} (r)}{2} + \frac{1}{2} \log_{3} (q) + \frac{1}{2} (- \log_{3} (2)) + \frac{1}{2} \cdot - 1$

Step 11.2

Combine $\frac{1}{2}$ and $\log_{3} (q)$ .

$\frac{\log_{3} (r)}{2} + \frac{\log_{3} (q)}{2} + \frac{1}{2} (- \log_{3} (2)) + \frac{1}{2} \cdot - 1$

Step 11.3

Combine $\frac{1}{2}$ and $\log_{3} (2)$ .

$\frac{\log_{3} (r)}{2} + \frac{\log_{3} (q)}{2} - \frac{\log_{3} (2)}{2} + \frac{1}{2} \cdot - 1$

Step 11.4

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

$\frac{\log_{3} (r)}{2} + \frac{\log_{3} (q)}{2} - \frac{\log_{3} (2)}{2} + \frac{- 1}{2}$

Step 12

Move the negative in front of the fraction.

$\frac{\log_{3} (r)}{2} + \frac{\log_{3} (q)}{2} - \frac{\log_{3} (2)}{2} - \frac{1}{2}$