Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Multiply $2$ by $4$ .

$\frac{8 \ln (x) + \ln (x - 2)}{2} - \ln (7 x + 3)$

Step 11

To write $- \ln (7 x + 3)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{8 \ln (x) + \ln (x - 2)}{2} - \ln (7 x + 3) \cdot \frac{2}{2}$

Step 12

Combine $- \ln (7 x + 3)$ and $\frac{2}{2}$ .

$\frac{8 \ln (x) + \ln (x - 2)}{2} + \frac{- \ln (7 x + 3) \cdot 2}{2}$

Step 13

Combine the numerators over the common denominator.

$\frac{8 \ln (x) + \ln (x - 2) - \ln (7 x + 3) \cdot 2}{2}$

Step 14

Multiply $2$ by $- 1$ .

$\frac{8 \ln (x) + \ln (x - 2) - 2 \ln (7 x + 3)}{2}$