Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

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

Step 7.2

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

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Step 7.2.1

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

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

Step 7.2.2

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

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

Step 7.3

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

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