Precalculus Examples

$\log (\sqrt[3]{x^{5} y^{2} z})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{5} y^{2} z}$ as ${(x^{5} y^{2} z)}^{\frac{1}{3}}$ .

$\log ({(x^{5} y^{2} z)}^{\frac{1}{3}})$

Expand $\log ({(x^{5} y^{2} z)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \log (x^{5} y^{2} z)$

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

$\frac{1}{3} (\log (x^{5} y^{2}) + \log (z))$

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

$\frac{1}{3} (\log (x^{5}) + \log (y^{2}) + \log (z))$

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

$\frac{1}{3} (5 \log (x) + \log (y^{2}) + \log (z))$

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

$\frac{1}{3} (5 \log (x) + 2 \log (y) + \log (z))$

Apply the distributive property.

$\frac{1}{3} (5 \log (x)) + \frac{1}{3} (2 \log (y)) + \frac{1}{3} \log (z)$

Simplify.

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Multiply $\frac{1}{3} (5 \log (x))$ .

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Combine $5$ and $\frac{1}{3}$ .

$\frac{5}{3} \log (x) + \frac{1}{3} (2 \log (y)) + \frac{1}{3} \log (z)$

Combine $\frac{5}{3}$ and $\log (x)$ .

$\frac{5 \log (x)}{3} + \frac{1}{3} (2 \log (y)) + \frac{1}{3} \log (z)$

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

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

$\frac{5 \log (x)}{3} + \frac{2}{3} \log (y) + \frac{1}{3} \log (z)$

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

$\frac{5 \log (x)}{3} + \frac{2 \log (y)}{3} + \frac{1}{3} \log (z)$

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

$\frac{5 \log (x)}{3} + \frac{2 \log (y)}{3} + \frac{\log (z)}{3}$