Precalculus Examples

$\ln (\frac{\sqrt{7}}{\sqrt[3]{3}})$

Step 1

Rewrite $\ln (\frac{\sqrt{7}}{\sqrt[3]{3}})$ as $\ln (\sqrt{7}) - \ln (\sqrt[3]{3})$ .

$\ln (\sqrt{7}) - \ln (\sqrt[3]{3})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\ln (7^{\frac{1}{2}}) - \ln (\sqrt[3]{3})$

Step 3

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

$\ln (7^{\frac{1}{2}}) - \ln (3^{\frac{1}{3}})$

Step 4

Expand $\ln (7^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \ln (7) - \ln (3^{\frac{1}{3}})$

Step 5

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

$\frac{1}{2} \ln (7) - (\frac{1}{3} \ln (3))$

Step 6

Combine $\frac{1}{2}$ and $\ln (7)$ .

$\frac{\ln (7)}{2} - (\frac{1}{3} \ln (3))$

Step 7

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

$\frac{\ln (7)}{2} - \frac{\ln (3)}{3}$