Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Move $2$ to the left of $\ln (x)$ .

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