Precalculus Examples

$\log_{3} (\frac{\sqrt[3]{5}}{z^{2} y})$

Step 1

Rewrite $\log_{3} (\frac{\sqrt[3]{5}}{z^{2} y})$ as $\log_{3} (\sqrt[3]{5}) - \log_{3} (z^{2} y)$ .

$\log_{3} (\sqrt[3]{5}) - \log_{3} (z^{2} y)$

Step 2

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

$\log_{3} (5^{\frac{1}{3}}) - \log_{3} (z^{2} y)$

Step 3

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

$\frac{1}{3} \log_{3} (5) - \log_{3} (z^{2} y)$

Step 4

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

$\frac{\log_{3} (5)}{3} - \log_{3} (z^{2} y)$

Step 5

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

$\frac{\log_{3} (5)}{3} - (\log_{3} (z^{2}) + \log_{3} (y))$

Step 6

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

$\frac{\log_{3} (5)}{3} - (2 \log_{3} (z) + \log_{3} (y))$

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

$\frac{\log_{3} (5)}{3} - (2 \log_{3} (z)) - \log_{3} (y)$

Step 7.2

Multiply $2$ by $- 1$ .

$\frac{\log_{3} (5)}{3} - 2 \log_{3} (z) - \log_{3} (y)$