Precalculus Examples

$\ln (\frac{x^{8} \sqrt{x^{2} + 8}}{{(x + 8)}^{2}})$

Step 1

Rewrite $\ln (\frac{x^{8} \sqrt{x^{2} + 8}}{{(x + 8)}^{2}})$ as $\ln (x^{8} \sqrt{x^{2} + 8}) - \ln ({(x + 8)}^{2})$ .

$\ln (x^{8} \sqrt{x^{2} + 8}) - \ln ({(x + 8)}^{2})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 8}$ as ${(x^{2} + 8)}^{\frac{1}{2}}$ .

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

Step 3

Expand $\ln ({(x + 8)}^{2})$ by moving $2$ outside the logarithm.

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

Step 4

Multiply $2$ by $- 1$ .

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

Step 5

Rewrite $\ln (x^{8} {(x^{2} + 8)}^{\frac{1}{2}})$ as $\ln (x^{8}) + \ln ({(x^{2} + 8)}^{\frac{1}{2}})$ .

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

Step 6

Expand $\ln (x^{8})$ by moving $8$ outside the logarithm.

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

Step 7

Expand $\ln ({(x^{2} + 8)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

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

Step 8

Combine $\frac{1}{2}$ and $\ln (x^{2} + 8)$ .

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

Step 9

To write $8 \ln (x)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 10

Combine $8 \ln (x)$ and $\frac{2}{2}$ .

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

Step 11

Combine the numerators over the common denominator.

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

Step 12

Multiply $2$ by $8$ .

$\frac{16 \ln (x) + \ln (x^{2} + 8)}{2} - 2 \ln (x + 8)$

Step 13

To write $- 2 \ln (x + 8)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{16 \ln (x) + \ln (x^{2} + 8)}{2} - 2 \ln (x + 8) \cdot \frac{2}{2}$

Step 14

Combine $- 2 \ln (x + 8)$ and $\frac{2}{2}$ .

$\frac{16 \ln (x) + \ln (x^{2} + 8)}{2} + \frac{- 2 \ln (x + 8) \cdot 2}{2}$

Step 15

Combine the numerators over the common denominator.

$\frac{16 \ln (x) + \ln (x^{2} + 8) - 2 \ln (x + 8) \cdot 2}{2}$

Step 16

Multiply $2$ by $- 2$ .

$\frac{16 \ln (x) + \ln (x^{2} + 8) - 4 \ln (x + 8)}{2}$