Precalculus Examples

$\log_{7} (\sqrt[3]{x \cdot y \cdot z})$

Step 1

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

$\log_{7} ({(x \cdot y \cdot z)}^{\frac{1}{3}})$

Step 2

Expand $\log_{7} ({(x \cdot y \cdot z)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \log_{7} (x \cdot y \cdot z)$

Step 3

Rewrite $\log_{7} (x \cdot y \cdot z)$ as $\log_{7} (x \cdot y) + \log_{7} (z)$ .

$\frac{1}{3} (\log_{7} (x \cdot y) + \log_{7} (z))$

Step 4

Rewrite $\log_{7} (x \cdot y)$ as $\log_{7} (x) + \log_{7} (y)$ .

$\frac{1}{3} (\log_{7} (x) + \log_{7} (y) + \log_{7} (z))$

Step 5

Apply the distributive property.

$\frac{1}{3} \log_{7} (x) + \frac{1}{3} \log_{7} (y) + \frac{1}{3} \log_{7} (z)$

Step 6

Simplify.

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Step 6.1

Combine $\frac{1}{3}$ and $\log_{7} (x)$ .

$\frac{\log_{7} (x)}{3} + \frac{1}{3} \log_{7} (y) + \frac{1}{3} \log_{7} (z)$

Step 6.2

Combine $\frac{1}{3}$ and $\log_{7} (y)$ .

$\frac{\log_{7} (x)}{3} + \frac{\log_{7} (y)}{3} + \frac{1}{3} \log_{7} (z)$

Step 6.3

Combine $\frac{1}{3}$ and $\log_{7} (z)$ .

$\frac{\log_{7} (x)}{3} + \frac{\log_{7} (y)}{3} + \frac{\log_{7} (z)}{3}$