Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Expand $\ln (x^{4})$ by moving $4$ outside the logarithm.

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

Step 7

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

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

Step 8

To write $4 \ln (x)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 9

Combine $4 \ln (x)$ and $\frac{2}{2}$ .

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

Step 10

Combine the numerators over the common denominator.

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

Step 11

Multiply $2$ by $4$ .

$2 \ln (x - 4) + \frac{8 \ln (x) - \ln (x^{2} + 1)}{2}$

Step 12

To write $2 \ln (x - 4)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 \ln (x - 4) \cdot \frac{2}{2} + \frac{8 \ln (x) - \ln (x^{2} + 1)}{2}$

Step 13

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

$\frac{2 \ln (x - 4) \cdot 2}{2} + \frac{8 \ln (x) - \ln (x^{2} + 1)}{2}$

Step 14

Combine the numerators over the common denominator.

$\frac{2 \ln (x - 4) \cdot 2 + 8 \ln (x) - \ln (x^{2} + 1)}{2}$

Step 15

Multiply $2$ by $2$ .

$\frac{4 \ln (x - 4) + 8 \ln (x) - \ln (x^{2} + 1)}{2}$