Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Multiply $2$ by $- 1$ .

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

Step 6

Rewrite $\log (x y)$ as $\log (x) + \log (y)$ .

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

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

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

Step 7.2

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

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

Step 7.3

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

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