Precalculus Examples

$\log (\frac{\sqrt[3]{a^{2} b}}{c})$

Step 1

Rewrite $\log (\frac{\sqrt[3]{a^{2} b}}{c})$ as $\log (\sqrt[3]{a^{2} b}) - \log (c)$ .

$\log (\sqrt[3]{a^{2} b}) - \log (c)$

Step 2

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

$\log ({(a^{2} b)}^{\frac{1}{3}}) - \log (c)$

Step 3

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

$\frac{1}{3} \log (a^{2} b) - \log (c)$

Step 4

Rewrite $\log (a^{2} b)$ as $\log (a^{2}) + \log (b)$ .

$\frac{1}{3} (\log (a^{2}) + \log (b)) - \log (c)$

Step 5

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

$\frac{1}{3} (2 \log (a) + \log (b)) - \log (c)$

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

$\frac{1}{3} (2 \log (a)) + \frac{1}{3} \log (b) - \log (c)$

Step 6.2

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

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Step 6.2.1

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

$\frac{2}{3} \log (a) + \frac{1}{3} \log (b) - \log (c)$

Step 6.2.2

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

$\frac{2 \log (a)}{3} + \frac{1}{3} \log (b) - \log (c)$

Step 6.3

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

$\frac{2 \log (a)}{3} + \frac{\log (b)}{3} - \log (c)$