Precalculus Examples

$\log_{3} (\frac{\sqrt[3]{4}}{r^{2} q})$

Step 1

Rewrite $\log_{3} (\frac{\sqrt[3]{4}}{r^{2} q})$ as $\log_{3} (\sqrt[3]{4}) - \log_{3} (r^{2} q)$ .

$\log_{3} (\sqrt[3]{4}) - \log_{3} (r^{2} q)$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$\log_{3} (4^{\frac{1}{3}}) - \log_{3} (r^{2} q)$

Step 3

Expand $\log_{3} (4^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \log_{3} (4) - \log_{3} (r^{2} q)$

Step 4

Rewrite $\log_{3} (4)$ as $\log_{3} (2^{2})$ .

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

Step 5

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

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

Step 6

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

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Step 6.1

Combine $2$ and $\frac{1}{3}$ .

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

Step 6.2

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify each term.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Multiply $2$ by $- 1$ .

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