Precalculus Examples

$\ln (\frac{\sqrt[3]{x}}{\sqrt[3]{y}})$

Step 1

Rewrite $\ln (\frac{\sqrt[3]{x}}{\sqrt[3]{y}})$ as $\ln (\sqrt[3]{x}) - \ln (\sqrt[3]{y})$ .

$\ln (\sqrt[3]{x}) - \ln (\sqrt[3]{y})$

Step 2

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

$\ln (x^{\frac{1}{3}}) - \ln (\sqrt[3]{y})$

Step 3

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

$\ln (x^{\frac{1}{3}}) - \ln (y^{\frac{1}{3}})$

Step 4

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

$\frac{1}{3} \ln (x) - \ln (y^{\frac{1}{3}})$

Step 5

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

$\frac{1}{3} \ln (x) - (\frac{1}{3} \ln (y))$

Step 6

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

$\frac{1}{3} \ln (x) - \frac{\ln (y)}{3}$

Step 7

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

$\frac{\ln (x)}{3} - \frac{\ln (y)}{3}$