Precalculus Examples

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

Step 1

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

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

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x - 2}$ as ${(3 x - 2)}^{\frac{1}{2}}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Multiply $2$ by $- 1$ .

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