Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Rewrite $\ln (x^{\frac{1}{2}} {(1 + x)}^{5})$ as $\ln (x^{\frac{1}{2}}) + \ln ({(1 + x)}^{5})$ .

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

Step 5

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

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

Step 6

Expand $\ln ({(1 + x)}^{5})$ by moving $5$ outside the logarithm.

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

Step 7

Simplify each term.

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Step 7.1

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

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

Step 7.2

To write $5 \ln (1 + x)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 7.3

Combine $5 \ln (1 + x)$ and $\frac{2}{2}$ .

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

Step 7.4

Combine the numerators over the common denominator.

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

Step 7.5

Multiply $2$ by $5$ .

$4 \ln (x) - \frac{\ln (x) + 10 \ln (1 + x)}{2}$