Precalculus Examples

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

Step 1

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

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

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{4} + 1}$ as ${(x^{4} + 1)}^{\frac{1}{2}}$ .

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

Step 3

Expand $\ln ({(x^{4} + 1)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

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

Step 4

Combine $\frac{1}{2}$ and $\ln (x^{4} + 1)$ .

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

Step 5

Rewrite $\ln (x^{4} \sin^{5} (x))$ as $\ln (x^{4}) + \ln (\sin^{5} (x))$ .

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Combine the numerators over the common denominator.

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

Step 11

Multiply $2$ by $4$ .

$5 \ln (\sin (x)) + \frac{8 \ln (x) - \ln (x^{4} + 1)}{2}$