Precalculus Examples

$\log (\sqrt[5]{x \sqrt{y z}})$

Step 1

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

$\log (\sqrt[5]{x {(y z)}^{\frac{1}{2}}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x {(y z)}^{\frac{1}{2}}}$ as ${(x {(y z)}^{\frac{1}{2}})}^{\frac{1}{5}}$ .

$\log ({(x {(y z)}^{\frac{1}{2}})}^{\frac{1}{5}})$

Step 3

Expand $\log ({(x {(y z)}^{\frac{1}{2}})}^{\frac{1}{5}})$ by moving $\frac{1}{5}$ outside the logarithm.

$\frac{1}{5} \log (x {(y z)}^{\frac{1}{2}})$

Step 4

Rewrite $\log (x {(y z)}^{\frac{1}{2}})$ as $\log (x) + \log ({(y z)}^{\frac{1}{2}})$ .

$\frac{1}{5} (\log (x) + \log ({(y z)}^{\frac{1}{2}}))$

Step 5

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

$\frac{1}{5} (\log (x) + \frac{1}{2} \log (y z))$

Step 6

Rewrite $\log (y z)$ as $\log (y) + \log (z)$ .

$\frac{1}{5} (\log (x) + \frac{1}{2} (\log (y) + \log (z)))$

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

$\frac{1}{5} (\log (x) + \frac{1}{2} \log (y) + \frac{1}{2} \log (z))$

Step 7.2

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

$\frac{1}{5} (\log (x) + \frac{\log (y)}{2} + \frac{1}{2} \log (z))$

Step 7.3

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

$\frac{1}{5} (\log (x) + \frac{\log (y)}{2} + \frac{\log (z)}{2})$

Step 8

Apply the distributive property.

$\frac{1}{5} \log (x) + \frac{1}{5} \cdot \frac{\log (y)}{2} + \frac{1}{5} \cdot \frac{\log (z)}{2}$

Step 9

Simplify.

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Step 9.1

Combine $\frac{1}{5}$ and $\log (x)$ .

$\frac{\log (x)}{5} + \frac{1}{5} \cdot \frac{\log (y)}{2} + \frac{1}{5} \cdot \frac{\log (z)}{2}$

Step 9.2

Multiply $\frac{1}{5} \cdot \frac{\log (y)}{2}$ .

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Step 9.2.1

Multiply $\frac{1}{5}$ by $\frac{\log (y)}{2}$ .

$\frac{\log (x)}{5} + \frac{\log (y)}{5 \cdot 2} + \frac{1}{5} \cdot \frac{\log (z)}{2}$

Step 9.2.2

Multiply $5$ by $2$ .

$\frac{\log (x)}{5} + \frac{\log (y)}{10} + \frac{1}{5} \cdot \frac{\log (z)}{2}$

Step 9.3

Multiply $\frac{1}{5} \cdot \frac{\log (z)}{2}$ .

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Step 9.3.1

Multiply $\frac{1}{5}$ by $\frac{\log (z)}{2}$ .

$\frac{\log (x)}{5} + \frac{\log (y)}{10} + \frac{\log (z)}{5 \cdot 2}$

Step 9.3.2

Multiply $5$ by $2$ .

$\frac{\log (x)}{5} + \frac{\log (y)}{10} + \frac{\log (z)}{10}$