Precalculus Examples

$\ln (\frac{x^{\frac{1}{2}}}{x^{2} \sqrt{1 + x^{2}}})$

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify each term.

Tap for more steps...

Step 7.1

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

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

Step 7.2

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

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

Step 7.3

To write $2 \ln (x)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 7.4

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

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

Step 7.5

Combine the numerators over the common denominator.

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

Step 7.6

Multiply $2$ by $2$ .

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