Precalculus Examples

$\log_{3} (x^{2} \cdot \sqrt{\frac{y}{z}})$

Step 1

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

$\log_{3} (x^{2}) + \log_{3} (\sqrt{\frac{y}{z}})$

Step 2

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

$\log_{3} (x^{2}) + \log_{3} ({(\frac{y}{z})}^{\frac{1}{2}})$

Step 3

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

$2 \log_{3} (x) + \log_{3} ({(\frac{y}{z})}^{\frac{1}{2}})$

Step 4

Expand $\log_{3} ({(\frac{y}{z})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$2 \log_{3} (x) + \frac{1}{2} \log_{3} (\frac{y}{z})$

Step 5

Rewrite $\log_{3} (\frac{y}{z})$ as $\log_{3} (y) - \log_{3} (z)$ .

$2 \log_{3} (x) + \frac{1}{2} (\log_{3} (y) - \log_{3} (z))$

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

$2 \log_{3} (x) + \frac{1}{2} \log_{3} (y) + \frac{1}{2} (- \log_{3} (z))$

Step 6.2

Combine $\frac{1}{2}$ and $\log_{3} (y)$ .

$2 \log_{3} (x) + \frac{\log_{3} (y)}{2} + \frac{1}{2} (- \log_{3} (z))$

Step 6.3

Combine $\frac{1}{2}$ and $\log_{3} (z)$ .

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