Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify each term.

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Step 5.1

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

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

Step 5.2

Apply the distributive property.

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