Calculus Examples

$\ln (\frac{7 \sqrt{5}}{\sqrt[3]{3}})$

Step 1

Rewrite $\ln (\frac{7 \sqrt{5}}{\sqrt[3]{3}})$ as $\ln (7 \sqrt{5}) - \ln (\sqrt[3]{3})$ .

$\ln (7 \sqrt{5}) - \ln (\sqrt[3]{3})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\ln (7 \cdot 5^{\frac{1}{2}}) - \ln (\sqrt[3]{3})$

Step 3

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

$\ln (7 \cdot 5^{\frac{1}{2}}) - \ln (3^{\frac{1}{3}})$

Step 4

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

$\ln (7 \cdot 5^{\frac{1}{2}}) - (\frac{1}{3} \ln (3))$

Step 5

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

$\ln (7 \cdot 5^{\frac{1}{2}}) - \frac{\ln (3)}{3}$

Step 6

Rewrite $\ln (7 \cdot 5^{\frac{1}{2}})$ as $\ln (7) + \ln (5^{\frac{1}{2}})$ .

$\ln (7) + \ln (5^{\frac{1}{2}}) - \frac{\ln (3)}{3}$

Step 7

Expand $\ln (5^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\ln (7) + \frac{1}{2} \ln (5) - \frac{\ln (3)}{3}$

Step 8

Combine $\frac{1}{2}$ and $\ln (5)$ .

$\ln (7) + \frac{\ln (5)}{2} - \frac{\ln (3)}{3}$