Precalculus Examples

$\log_{3} (x^{14} \cdot \sqrt[3]{y^{8}})$

Step 1

Rewrite $\log_{3} (x^{14} \cdot \sqrt[3]{y^{8}})$ as $\log_{3} (x^{14}) + \log_{3} (\sqrt[3]{y^{8}})$ .

$\log_{3} (x^{14}) + \log_{3} (\sqrt[3]{y^{8}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y^{8}}$ as $y^{\frac{8}{3}}$ .

$\log_{3} (x^{14}) + \log_{3} (y^{\frac{8}{3}})$

Step 3

Expand $\log_{3} (x^{14})$ by moving $14$ outside the logarithm.

$14 \log_{3} (x) + \log_{3} (y^{\frac{8}{3}})$

Step 4

Expand $\log_{3} (y^{\frac{8}{3}})$ by moving $\frac{8}{3}$ outside the logarithm.

$14 \log_{3} (x) + \frac{8}{3} \log_{3} (y)$

Step 5

Combine $\frac{8}{3}$ and $\log_{3} (y)$ .

$14 \log_{3} (x) + \frac{8 \log_{3} (y)}{3}$