Precalculus Examples

$\ln (\frac{\sqrt{x^{2} + 4}}{y \cdot {(x - 1)}^{6}})$

Step 1

Rewrite $\ln (\frac{\sqrt{x^{2} + 4}}{y \cdot {(x - 1)}^{6}})$ as $\ln (\sqrt{x^{2} + 4}) - \ln (y \cdot {(x - 1)}^{6})$ .

$\ln (\sqrt{x^{2} + 4}) - \ln (y \cdot {(x - 1)}^{6})$

Step 2

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

$\ln ({(x^{2} + 4)}^{\frac{1}{2}}) - \ln (y \cdot {(x - 1)}^{6})$

Step 3

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

$\frac{1}{2} \ln (x^{2} + 4) - \ln (y \cdot {(x - 1)}^{6})$

Step 4

Rewrite $\ln (y \cdot {(x - 1)}^{6})$ as $\ln (y) + \ln ({(x - 1)}^{6})$ .

$\frac{1}{2} \ln (x^{2} + 4) - (\ln (y) + \ln ({(x - 1)}^{6}))$

Step 5

Expand $\ln ({(x - 1)}^{6})$ by moving $6$ outside the logarithm.

$\frac{1}{2} \ln (x^{2} + 4) - (\ln (y) + 6 \ln (x - 1))$

Step 6

Simplify each term.

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Step 6.1

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

$\frac{\ln (x^{2} + 4)}{2} - (\ln (y) + 6 \ln (x - 1))$

Step 6.2

Apply the distributive property.

$\frac{\ln (x^{2} + 4)}{2} - \ln (y) - (6 \ln (x - 1))$

Step 6.3

Multiply $6$ by $- 1$ .

$\frac{\ln (x^{2} + 4)}{2} - \ln (y) - 6 \ln (x - 1)$

Step 7

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

$\frac{\ln (x^{2} + 4)}{2} - \ln (y) \cdot \frac{2}{2} - 6 \ln (x - 1)$

Step 8

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

$\frac{\ln (x^{2} + 4)}{2} + \frac{- \ln (y) \cdot 2}{2} - 6 \ln (x - 1)$

Step 9

Combine the numerators over the common denominator.

$\frac{\ln (x^{2} + 4) - \ln (y) \cdot 2}{2} - 6 \ln (x - 1)$

Step 10

Multiply $2$ by $- 1$ .

$\frac{\ln (x^{2} + 4) - 2 \ln (y)}{2} - 6 \ln (x - 1)$

Step 11

To write $- 6 \ln (x - 1)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\ln (x^{2} + 4) - 2 \ln (y)}{2} - 6 \ln (x - 1) \cdot \frac{2}{2}$

Step 12

Combine $- 6 \ln (x - 1)$ and $\frac{2}{2}$ .

$\frac{\ln (x^{2} + 4) - 2 \ln (y)}{2} + \frac{- 6 \ln (x - 1) \cdot 2}{2}$

Step 13

Combine the numerators over the common denominator.

$\frac{\ln (x^{2} + 4) - 2 \ln (y) - 6 \ln (x - 1) \cdot 2}{2}$

Step 14

Multiply $2$ by $- 6$ .

$\frac{\ln (x^{2} + 4) - 2 \ln (y) - 12 \ln (x - 1)}{2}$