Precalculus Examples

$\log (\sqrt[3]{x y^{4} z^{2}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x y^{4} z^{2}}$ as ${(x y^{4} z^{2})}^{\frac{1}{3}}$ .

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

Step 2

Expand $\log ({(x y^{4} z^{2})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

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

Step 3

Rewrite $\log (x y^{4} z^{2})$ as $\log (x y^{4}) + \log (z^{2})$ .

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

Step 4

Rewrite $\log (x y^{4})$ as $\log (x) + \log (y^{4})$ .

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

Step 5

Expand $\log (y^{4})$ by moving $4$ outside the logarithm.

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

Step 6

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

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

Step 7

Apply the distributive property.

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

Step 8

Simplify.

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Step 8.1

Combine $\frac{1}{3}$ and $\log (x)$ .

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

Step 8.2

Multiply $\frac{1}{3} (4 \log (y))$ .

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Step 8.2.1

Combine $4$ and $\frac{1}{3}$ .

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

Step 8.2.2

Combine $\frac{4}{3}$ and $\log (y)$ .

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

Step 8.3

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

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Step 8.3.1

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

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

Step 8.3.2

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

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