Precalculus Examples

$\log_{3} (\frac{\sqrt{x z}}{y^{5}})$

Step 1

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

$\log_{3} (\sqrt{x z}) - \log_{3} (y^{5})$

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Multiply $5$ by $- 1$ .

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

Step 6

Rewrite $\log_{3} (x z)$ as $\log_{3} (x) + \log_{3} (z)$ .

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

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

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

Step 7.2

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

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

Step 7.3

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

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