Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

To write $2 \ln (x^{2} + 1)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 7

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

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

Multiply $2$ by $2$ .

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