Precalculus Examples

$\log_{3} (\frac{p^{2} q}{\sqrt[5]{3 q - 1}})$

Step 1

Rewrite $\log_{3} (\frac{p^{2} q}{\sqrt[5]{3 q - 1}})$ as $\log_{3} (p^{2} q) - \log_{3} (\sqrt[5]{3 q - 1})$ .

$\log_{3} (p^{2} q) - \log_{3} (\sqrt[5]{3 q - 1})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{3 q - 1}$ as ${(3 q - 1)}^{\frac{1}{5}}$ .

$\log_{3} (p^{2} q) - \log_{3} ({(3 q - 1)}^{\frac{1}{5}})$

Step 3

Expand $\log_{3} ({(3 q - 1)}^{\frac{1}{5}})$ by moving $\frac{1}{5}$ outside the logarithm.

$\log_{3} (p^{2} q) - (\frac{1}{5} \log_{3} (3 q - 1))$

Step 4

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

$\log_{3} (p^{2} q) - \frac{\log_{3} (3 q - 1)}{5}$

Step 5

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

$\log_{3} (p^{2}) + \log_{3} (q) - \frac{\log_{3} (3 q - 1)}{5}$

Step 6

Expand $\log_{3} (p^{2})$ by moving $2$ outside the logarithm.

$2 \log_{3} (p) + \log_{3} (q) - \frac{\log_{3} (3 q - 1)}{5}$

Step 7

To write $2 \log_{3} (p)$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$2 \log_{3} (p) \cdot \frac{5}{5} - \frac{\log_{3} (3 q - 1)}{5} + \log_{3} (q)$

Step 8

Combine $2 \log_{3} (p)$ and $\frac{5}{5}$ .

$\frac{2 \log_{3} (p) \cdot 5}{5} - \frac{\log_{3} (3 q - 1)}{5} + \log_{3} (q)$

Step 9

Combine the numerators over the common denominator.

$\frac{2 \log_{3} (p) \cdot 5 - \log_{3} (3 q - 1)}{5} + \log_{3} (q)$

Step 10

Multiply $5$ by $2$ .

$\frac{10 \log_{3} (p) - \log_{3} (3 q - 1)}{5} + \log_{3} (q)$