Precalculus Examples

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

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x \cdot y \cdot z}$ as ${(x \cdot y \cdot z)}^{\frac{1}{2}}$ .

$\log_{9} ({(x \cdot y \cdot z)}^{\frac{1}{2}})$

Step 2

Expand $\log_{9} ({(x \cdot y \cdot z)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{9} (x \cdot y \cdot z)$

Step 3

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

$\frac{1}{2} (\log_{9} (x \cdot y) + \log_{9} (z))$

Step 4

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

$\frac{1}{2} (\log_{9} (x) + \log_{9} (y) + \log_{9} (z))$

Step 5

Apply the distributive property.

$\frac{1}{2} \log_{9} (x) + \frac{1}{2} \log_{9} (y) + \frac{1}{2} \log_{9} (z)$

Step 6

Simplify.

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Step 6.1

Combine $\frac{1}{2}$ and $\log_{9} (x)$ .

$\frac{\log_{9} (x)}{2} + \frac{1}{2} \log_{9} (y) + \frac{1}{2} \log_{9} (z)$

Step 6.2

Combine $\frac{1}{2}$ and $\log_{9} (y)$ .

$\frac{\log_{9} (x)}{2} + \frac{\log_{9} (y)}{2} + \frac{1}{2} \log_{9} (z)$

Step 6.3

Combine $\frac{1}{2}$ and $\log_{9} (z)$ .

$\frac{\log_{9} (x)}{2} + \frac{\log_{9} (y)}{2} + \frac{\log_{9} (z)}{2}$