Precalculus Examples

$\log (\sqrt{x^{3} \sqrt{y^{3} \sqrt{z}}})$

Step 1

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

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

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y^{3} z^{\frac{1}{2}}}$ as ${(y^{3} z^{\frac{1}{2}})}^{\frac{1}{2}}$ .

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

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3} {(y^{3} z^{\frac{1}{2}})}^{\frac{1}{2}}}$ as ${(x^{3} {(y^{3} z^{\frac{1}{2}})}^{\frac{1}{2}})}^{\frac{1}{2}}$ .

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

Step 4

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

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

Step 5

Rewrite $\log (x^{3} {(y^{3} z^{\frac{1}{2}})}^{\frac{1}{2}})$ as $\log (x^{3}) + \log ({(y^{3} z^{\frac{1}{2}})}^{\frac{1}{2}})$ .

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

Step 6

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

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

Step 7

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

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

Step 8

Rewrite $\log (y^{3} z^{\frac{1}{2}})$ as $\log (y^{3}) + \log (z^{\frac{1}{2}})$ .

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify each term.

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Step 11.1

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

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

Step 11.2

Apply the distributive property.

$\frac{1}{2} (3 \log (x) + \frac{1}{2} (3 \log (y)) + \frac{1}{2} \cdot \frac{\log (z)}{2})$

Step 11.3

Multiply $\frac{1}{2} (3 \log (y))$ .

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Step 11.3.1

Combine $3$ and $\frac{1}{2}$ .

$\frac{1}{2} (3 \log (x) + \frac{3}{2} \log (y) + \frac{1}{2} \cdot \frac{\log (z)}{2})$

Step 11.3.2

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

$\frac{1}{2} (3 \log (x) + \frac{3 \log (y)}{2} + \frac{1}{2} \cdot \frac{\log (z)}{2})$

Step 11.4

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

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Step 11.4.1

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

$\frac{1}{2} (3 \log (x) + \frac{3 \log (y)}{2} + \frac{\log (z)}{2 \cdot 2})$

Step 11.4.2

Multiply $2$ by $2$ .

$\frac{1}{2} (3 \log (x) + \frac{3 \log (y)}{2} + \frac{\log (z)}{4})$

Step 12

Apply the distributive property.

$\frac{1}{2} (3 \log (x)) + \frac{1}{2} \cdot \frac{3 \log (y)}{2} + \frac{1}{2} \cdot \frac{\log (z)}{4}$

Step 13

Simplify.

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Step 13.1

Multiply $\frac{1}{2} (3 \log (x))$ .

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Step 13.1.1

Combine $3$ and $\frac{1}{2}$ .

$\frac{3}{2} \log (x) + \frac{1}{2} \cdot \frac{3 \log (y)}{2} + \frac{1}{2} \cdot \frac{\log (z)}{4}$

Step 13.1.2

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

$\frac{3 \log (x)}{2} + \frac{1}{2} \cdot \frac{3 \log (y)}{2} + \frac{1}{2} \cdot \frac{\log (z)}{4}$

Step 13.2

Multiply $\frac{1}{2} \cdot \frac{3 \log (y)}{2}$ .

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Step 13.2.1

Multiply $\frac{1}{2}$ by $\frac{3 \log (y)}{2}$ .

$\frac{3 \log (x)}{2} + \frac{3 \log (y)}{2 \cdot 2} + \frac{1}{2} \cdot \frac{\log (z)}{4}$

Step 13.2.2

Multiply $2$ by $2$ .

$\frac{3 \log (x)}{2} + \frac{3 \log (y)}{4} + \frac{1}{2} \cdot \frac{\log (z)}{4}$

Step 13.3

Multiply $\frac{1}{2} \cdot \frac{\log (z)}{4}$ .

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Step 13.3.1

Multiply $\frac{1}{2}$ by $\frac{\log (z)}{4}$ .

$\frac{3 \log (x)}{2} + \frac{3 \log (y)}{4} + \frac{\log (z)}{2 \cdot 4}$

Step 13.3.2

Multiply $2$ by $4$ .

$\frac{3 \log (x)}{2} + \frac{3 \log (y)}{4} + \frac{\log (z)}{8}$