Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

Multiply $2$ by $2$ .

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

Step 10

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

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

Step 11

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

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

Step 12

Combine the numerators over the common denominator.

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

Step 13

Move $2$ to the left of $\ln (2 x - 1)$ .

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