Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

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

Apply the distributive property.

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

Step 7

Simplify.

Tap for more steps...

Step 7.1

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

Tap for more steps...

Step 7.1.1

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

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

Step 7.1.2

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

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

Step 7.2

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

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

Step 7.3

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

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