Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Rewrite $\log (3 z)$ as $\log (3) + \log (z)$ .

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

Step 8

Simplify each term.

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Step 8.1

Apply the distributive property.

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

Step 8.2

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

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Step 8.2.1

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

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

Step 8.2.2

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

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

Step 8.3

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

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Step 8.3.1

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

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

Step 8.3.2

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

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

Step 8.4

Apply the distributive property.

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