Precalculus Examples

$\log_{9} (z \sqrt[3]{x \cdot y})$

Step 1

Rewrite $\log_{9} (z \sqrt[3]{x \cdot y})$ as $\log_{9} (z) + \log_{9} (\sqrt[3]{x \cdot y})$ .

$\log_{9} (z) + \log_{9} (\sqrt[3]{x \cdot y})$

Step 2

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

$\log_{9} (z) + \log_{9} ({(x \cdot y)}^{\frac{1}{3}})$

Step 3

Expand $\log_{9} ({(x \cdot y)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\log_{9} (z) + \frac{1}{3} \log_{9} (x \cdot y)$

Step 4

Rewrite $\log_{9} (x \cdot y)$ as $\log_{9} (x) + \log_{9} (y)$ .

$\log_{9} (z) + \frac{1}{3} (\log_{9} (x) + \log_{9} (y))$

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

$\log_{9} (z) + \frac{1}{3} \log_{9} (x) + \frac{1}{3} \log_{9} (y)$

Step 5.2

Combine $\frac{1}{3}$ and $\log_{9} (x)$ .

$\log_{9} (z) + \frac{\log_{9} (x)}{3} + \frac{1}{3} \log_{9} (y)$

Step 5.3

Combine $\frac{1}{3}$ and $\log_{9} (y)$ .

$\log_{9} (z) + \frac{\log_{9} (x)}{3} + \frac{\log_{9} (y)}{3}$