Precalculus Examples

$\log (\sqrt{x \sqrt{y} \sqrt{z}})$

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify each term.

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Step 9.1

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

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

Step 9.2

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

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

Step 10

Apply the distributive property.

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

Step 11

Simplify.

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Step 11.1

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

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

Step 11.2

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

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Step 11.2.1

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

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

Step 11.2.2

Multiply $2$ by $2$ .

$\frac{\log (x)}{2} + \frac{\log (y)}{4} + \frac{1}{2} \cdot \frac{\log (z)}{2}$

Step 11.3

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

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Step 11.3.1

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

$\frac{\log (x)}{2} + \frac{\log (y)}{4} + \frac{\log (z)}{2 \cdot 2}$

Step 11.3.2

Multiply $2$ by $2$ .

$\frac{\log (x)}{2} + \frac{\log (y)}{4} + \frac{\log (z)}{4}$