Precalculus Examples

$\ln (\frac{(4 x^{5} - x - 1) \sqrt{x - 7}}{{(x^{2} + 1)}^{3}})$

Step 1

Rewrite $\ln (\frac{(4 x^{5} - x - 1) \sqrt{x - 7}}{{(x^{2} + 1)}^{3}})$ as $\ln ((4 x^{5} - x - 1) \sqrt{x - 7}) - \ln ({(x^{2} + 1)}^{3})$ .

$\ln ((4 x^{5} - x - 1) \sqrt{x - 7}) - \ln ({(x^{2} + 1)}^{3})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 7}$ as ${(x - 7)}^{\frac{1}{2}}$ .

$\ln ((4 x^{5} - x - 1) {(x - 7)}^{\frac{1}{2}}) - \ln ({(x^{2} + 1)}^{3})$

Step 3

Expand $\ln ({(x^{2} + 1)}^{3})$ by moving $3$ outside the logarithm.

$\ln ((4 x^{5} - x - 1) {(x - 7)}^{\frac{1}{2}}) - (3 \ln (x^{2} + 1))$

Step 4

Multiply $3$ by $- 1$ .

$\ln ((4 x^{5} - x - 1) {(x - 7)}^{\frac{1}{2}}) - 3 \ln (x^{2} + 1)$

Step 5

Rewrite $\ln ((4 x^{5} - x - 1) {(x - 7)}^{\frac{1}{2}})$ as $\ln (4 x^{5} - x - 1) + \ln ({(x - 7)}^{\frac{1}{2}})$ .

$\ln (4 x^{5} - x - 1) + \ln ({(x - 7)}^{\frac{1}{2}}) - 3 \ln (x^{2} + 1)$

Step 6

Expand $\ln ({(x - 7)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\ln (4 x^{5} - x - 1) + \frac{1}{2} \ln (x - 7) - 3 \ln (x^{2} + 1)$

Step 7

Combine $\frac{1}{2}$ and $\ln (x - 7)$ .

$\ln (4 x^{5} - x - 1) + \frac{\ln (x - 7)}{2} - 3 \ln (x^{2} + 1)$

Step 8

To write $\ln (4 x^{5} - x - 1)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\ln (4 x^{5} - x - 1) \cdot \frac{2}{2} + \frac{\ln (x - 7)}{2} - 3 \ln (x^{2} + 1)$

Step 9

Combine $\ln (4 x^{5} - x - 1)$ and $\frac{2}{2}$ .

$\frac{\ln (4 x^{5} - x - 1) \cdot 2}{2} + \frac{\ln (x - 7)}{2} - 3 \ln (x^{2} + 1)$

Step 10

Combine the numerators over the common denominator.

$\frac{\ln (4 x^{5} - x - 1) \cdot 2 + \ln (x - 7)}{2} - 3 \ln (x^{2} + 1)$

Step 11

Move $2$ to the left of $\ln (4 x^{5} - x - 1)$ .

$\frac{2 \ln (4 x^{5} - x - 1) + \ln (x - 7)}{2} - 3 \ln (x^{2} + 1)$

Step 12

To write $- 3 \ln (x^{2} + 1)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{2 \ln (4 x^{5} - x - 1) + \ln (x - 7)}{2} - 3 \ln (x^{2} + 1) \cdot \frac{2}{2}$

Step 13

Combine $- 3 \ln (x^{2} + 1)$ and $\frac{2}{2}$ .

$\frac{2 \ln (4 x^{5} - x - 1) + \ln (x - 7)}{2} + \frac{- 3 \ln (x^{2} + 1) \cdot 2}{2}$

Step 14

Combine the numerators over the common denominator.

$\frac{2 \ln (4 x^{5} - x - 1) + \ln (x - 7) - 3 \ln (x^{2} + 1) \cdot 2}{2}$

Step 15

Multiply $2$ by $- 3$ .

$\frac{2 \ln (4 x^{5} - x - 1) + \ln (x - 7) - 6 \ln (x^{2} + 1)}{2}$