Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Rewrite $\log (y^{5} z^{\frac{5}{2}})$ as $\log (y^{5}) + \log (z^{\frac{5}{2}})$ .

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

Step 8

Expand $\log (y^{5})$ by moving $5$ outside the logarithm.

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

Step 9

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

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

Step 10

Simplify each term.

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Step 10.1

Combine $\frac{5}{2}$ and $\log (z)$ .

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

Step 10.2

Apply the distributive property.

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

Step 10.3

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

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Step 10.3.1

Combine $5$ and $\frac{1}{2}$ .

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

Step 10.3.2

Combine $\frac{5}{2}$ and $\log (y)$ .

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

Step 10.4

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

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Step 10.4.1

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

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

Step 10.4.2

Multiply $2$ by $2$ .

$\frac{1}{2} (\log (x) + \frac{5 \log (y)}{2} + \frac{5 \log (z)}{4})$

Step 11

Apply the distributive property.

$\frac{1}{2} \log (x) + \frac{1}{2} \cdot \frac{5 \log (y)}{2} + \frac{1}{2} \cdot \frac{5 \log (z)}{4}$

Step 12

Simplify.

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Step 12.1

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

$\frac{\log (x)}{2} + \frac{1}{2} \cdot \frac{5 \log (y)}{2} + \frac{1}{2} \cdot \frac{5 \log (z)}{4}$

Step 12.2

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

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Step 12.2.1

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

$\frac{\log (x)}{2} + \frac{5 \log (y)}{2 \cdot 2} + \frac{1}{2} \cdot \frac{5 \log (z)}{4}$

Step 12.2.2

Multiply $2$ by $2$ .

$\frac{\log (x)}{2} + \frac{5 \log (y)}{4} + \frac{1}{2} \cdot \frac{5 \log (z)}{4}$

Step 12.3

Multiply $\frac{1}{2} \cdot \frac{5 \log (z)}{4}$ .

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Step 12.3.1

Multiply $\frac{1}{2}$ by $\frac{5 \log (z)}{4}$ .

$\frac{\log (x)}{2} + \frac{5 \log (y)}{4} + \frac{5 \log (z)}{2 \cdot 4}$

Step 12.3.2

Multiply $2$ by $4$ .

$\frac{\log (x)}{2} + \frac{5 \log (y)}{4} + \frac{5 \log (z)}{8}$