Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Multiply $4$ by $- 1$ .

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

Step 5

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

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

Step 6

Expand $\ln ({(x - 2)}^{5})$ by moving $5$ outside the logarithm.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Combine the numerators over the common denominator.

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

Step 12

Multiply $2$ by $5$ .

$\frac{10 \ln (x - 2) + \ln (x + 1)}{2} - 4 \ln (x)$