Precalculus Examples

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

Step 1

Rewrite $\ln (\frac{\sqrt{x^{2} + 1}}{x^{4} {(2 x - 1)}^{3}})$ as $\ln (\sqrt{x^{2} + 1}) - \ln (x^{4} {(2 x - 1)}^{3})$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Rewrite $\ln (x^{4} {(2 x - 1)}^{3})$ as $\ln (x^{4}) + \ln ({(2 x - 1)}^{3})$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify each term.

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Step 7.1

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

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Multiply $4$ by $- 1$ .

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

Step 7.4

Multiply $3$ by $- 1$ .

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

Step 8

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

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

Step 9

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

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

Step 10

Combine the numerators over the common denominator.

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

Step 11

Multiply $2$ by $- 4$ .

$\frac{\ln (x^{2} + 1) - 8 \ln (x)}{2} - 3 \ln (2 x - 1)$

Step 12

To write $- 3 \ln (2 x - 1)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\ln (x^{2} + 1) - 8 \ln (x)}{2} - 3 \ln (2 x - 1) \cdot \frac{2}{2}$

Step 13

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

$\frac{\ln (x^{2} + 1) - 8 \ln (x)}{2} + \frac{- 3 \ln (2 x - 1) \cdot 2}{2}$

Step 14

Combine the numerators over the common denominator.

$\frac{\ln (x^{2} + 1) - 8 \ln (x) - 3 \ln (2 x - 1) \cdot 2}{2}$

Step 15

Multiply $2$ by $- 3$ .

$\frac{\ln (x^{2} + 1) - 8 \ln (x) - 6 \ln (2 x - 1)}{2}$