Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Apply the distributive property.

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

Step 7

Apply the distributive property.

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

Step 8

Simplify.

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Step 8.1

Cancel the common factor of $2$ .

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Step 8.1.1

Factor $2$ out of $2 \log_{3} (x)$ .

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

Step 8.1.2

Cancel the common factor.

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

Step 8.1.3

Rewrite the expression.

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

Step 8.2

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

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

Step 8.3

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

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