Precalculus Examples

$\log_{3} (\frac{9 a^{3}}{\sqrt[5]{b}})$

Step 1

Rewrite $\log_{3} (\frac{9 a^{3}}{\sqrt[5]{b}})$ as $\log_{3} (9 a^{3}) - \log_{3} (\sqrt[5]{b})$ .

$\log_{3} (9 a^{3}) - \log_{3} (\sqrt[5]{b})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{b}$ as $b^{\frac{1}{5}}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Rewrite $\log_{3} (9 a^{3})$ as $\log_{3} (9) + \log_{3} (a^{3})$ .

$\log_{3} (9) + \log_{3} (a^{3}) - \frac{\log_{3} (b)}{5}$

Step 6

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

$\log_{3} (9) + 3 \log_{3} (a) - \frac{\log_{3} (b)}{5}$

Step 7

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

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