Precalculus Examples

$20 \log (\sqrt{x \sqrt{z}})$

Step 1

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

$20 \log (\sqrt{x z^{\frac{1}{2}}})$

Step 2

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

$20 \log ({(x z^{\frac{1}{2}})}^{\frac{1}{2}})$

Step 3

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

$20 (\frac{1}{2} \log (x z^{\frac{1}{2}}))$

Step 4

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

$20 (\frac{1}{2} (\log (x) + \log (z^{\frac{1}{2}})))$

Step 5

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

$20 (\frac{1}{2} (\log (x) + \frac{1}{2} \log (z)))$

Step 6

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

$20 (\frac{1}{2} (\log (x) + \frac{\log (z)}{2}))$

Step 7

Apply the distributive property.

$20 (\frac{1}{2} \log (x) + \frac{1}{2} \cdot \frac{\log (z)}{2})$

Step 8

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

$20 (\frac{\log (x)}{2} + \frac{1}{2} \cdot \frac{\log (z)}{2})$

Step 9

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

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Step 9.1

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

$20 (\frac{\log (x)}{2} + \frac{\log (z)}{2 \cdot 2})$

Step 9.2

Multiply $2$ by $2$ .

$20 (\frac{\log (x)}{2} + \frac{\log (z)}{4})$

Step 10

Apply the distributive property.

$20 \frac{\log (x)}{2} + 20 \frac{\log (z)}{4}$

Step 11

Cancel the common factor of $2$ .

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Step 11.1

Factor $2$ out of $20$ .

$2 (10) \frac{\log (x)}{2} + 20 \frac{\log (z)}{4}$

Step 11.2

Cancel the common factor.

$2 \cdot 10 \frac{\log (x)}{2} + 20 \frac{\log (z)}{4}$

Step 11.3

Rewrite the expression.

$10 \log (x) + 20 \frac{\log (z)}{4}$

Step 12

Cancel the common factor of $4$ .

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Step 12.1

Factor $4$ out of $20$ .

$10 \log (x) + 4 (5) \frac{\log (z)}{4}$

Step 12.2

Cancel the common factor.

$10 \log (x) + 4 \cdot 5 \frac{\log (z)}{4}$

Step 12.3

Rewrite the expression.

$10 \log (x) + 5 \log (z)$