Calculus Examples

$\ln (\sqrt[5]{\frac{4 x^{2} - 1}{4 x^{2} + 1}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{\frac{4 x^{2} - 1}{4 x^{2} + 1}}$ as ${(\frac{4 x^{2} - 1}{4 x^{2} + 1})}^{\frac{1}{5}}$ .

$\ln ({(\frac{4 x^{2} - 1}{4 x^{2} + 1})}^{\frac{1}{5}})$

Step 2

Expand $\ln ({(\frac{4 x^{2} - 1}{4 x^{2} + 1})}^{\frac{1}{5}})$ by moving $\frac{1}{5}$ outside the logarithm.

$\frac{1}{5} \ln (\frac{4 x^{2} - 1}{4 x^{2} + 1})$

Step 3

Rewrite $\ln (\frac{4 x^{2} - 1}{4 x^{2} + 1})$ as $\ln (4 x^{2} - 1) - \ln (4 x^{2} + 1)$ .

$\frac{1}{5} (\ln (4 x^{2} - 1) - \ln (4 x^{2} + 1))$

Step 4

Apply the distributive property.

$\frac{1}{5} \ln (4 x^{2} - 1) + \frac{1}{5} (- \ln (4 x^{2} + 1))$

Step 5

Combine $\frac{1}{5}$ and $\ln (4 x^{2} - 1)$ .

$\frac{\ln (4 x^{2} - 1)}{5} + \frac{1}{5} (- \ln (4 x^{2} + 1))$

Step 6

Combine $\frac{1}{5}$ and $\ln (4 x^{2} + 1)$ .

$\frac{\ln (4 x^{2} - 1)}{5} - \frac{\ln (4 x^{2} + 1)}{5}$

Step 7

Combine the numerators over the common denominator.

$\frac{\ln (4 x^{2} - 1) - \ln (4 x^{2} + 1)}{5}$