Precalculus Examples

$\ln (\sqrt[3]{u \cdot v \cdot w})$

Step 1

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

$\ln ({(u \cdot v \cdot w)}^{\frac{1}{3}})$

Step 2

Expand $\ln ({(u \cdot v \cdot w)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \ln (u \cdot v \cdot w)$

Step 3

Rewrite $\ln (u \cdot v \cdot w)$ as $\ln (u \cdot v) + \ln (w)$ .

$\frac{1}{3} (\ln (u \cdot v) + \ln (w))$

Step 4

Rewrite $\ln (u \cdot v)$ as $\ln (u) + \ln (v)$ .

$\frac{1}{3} (\ln (u) + \ln (v) + \ln (w))$

Step 5

Apply the distributive property.

$\frac{1}{3} \ln (u) + \frac{1}{3} \ln (v) + \frac{1}{3} \ln (w)$

Step 6

Simplify.

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Step 6.1

Combine $\frac{1}{3}$ and $\ln (u)$ .

$\frac{\ln (u)}{3} + \frac{1}{3} \ln (v) + \frac{1}{3} \ln (w)$

Step 6.2

Combine $\frac{1}{3}$ and $\ln (v)$ .

$\frac{\ln (u)}{3} + \frac{\ln (v)}{3} + \frac{1}{3} \ln (w)$

Step 6.3

Combine $\frac{1}{3}$ and $\ln (w)$ .

$\frac{\ln (u)}{3} + \frac{\ln (v)}{3} + \frac{\ln (w)}{3}$