Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Divide $2$ by $2$ .

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

Step 4

Rewrite $\ln (x^{2} \sin^{3} (x))$ as $\ln (x^{2}) + \ln (\sin^{3} (x))$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify.

$2 \ln (x) + 3 \ln (\sin (x)) - \ln (x + 1)$