Precalculus Examples

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

Step 1

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

$\ln (6 x \sqrt{1 + 2 x}) - \ln (x - 6)$

Step 2

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

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

Step 3

Rewrite $\ln (6 x {(1 + 2 x)}^{\frac{1}{2}})$ as $\ln (6 x) + \ln ({(1 + 2 x)}^{\frac{1}{2}})$ .

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

Step 4

Rewrite $\ln (6 x)$ as $\ln (6) + \ln (x)$ .

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

Step 5

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

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

Step 6

Simplify each term.

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Step 6.1

Rewrite $\ln (6)$ as $\ln (2 (3))$ .

$\ln (2 (3)) + \ln (x) + \frac{1}{2} \ln (1 + 2 x) - \ln (x - 6)$

Step 6.2

Rewrite $\ln (2 (3))$ as $\ln (2) + \ln (3)$ .

$\ln (2) + \ln (3) + \ln (x) + \frac{1}{2} \ln (1 + 2 x) - \ln (x - 6)$

Step 6.3

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

$\ln (2) + \ln (3) + \ln (x) + \frac{\ln (1 + 2 x)}{2} - \ln (x - 6)$

Step 7

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

$\ln (2) \cdot \frac{2}{2} + \frac{\ln (1 + 2 x)}{2} + \ln (3) + \ln (x) - \ln (x - 6)$

Step 8

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

$\frac{\ln (2) \cdot 2}{2} + \frac{\ln (1 + 2 x)}{2} + \ln (3) + \ln (x) - \ln (x - 6)$

Step 9

Combine the numerators over the common denominator.

$\frac{\ln (2) \cdot 2 + \ln (1 + 2 x)}{2} + \ln (3) + \ln (x) - \ln (x - 6)$

Step 10

Move $2$ to the left of $\ln (2)$ .

$\frac{2 \ln (2) + \ln (1 + 2 x)}{2} + \ln (3) + \ln (x) - \ln (x - 6)$

Step 11

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

$\frac{2 \ln (2) + \ln (1 + 2 x)}{2} - \ln (x - 6) \cdot \frac{2}{2} + \ln (3) + \ln (x)$

Step 12

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

$\frac{2 \ln (2) + \ln (1 + 2 x)}{2} + \frac{- \ln (x - 6) \cdot 2}{2} + \ln (3) + \ln (x)$

Step 13

Combine the numerators over the common denominator.

$\frac{2 \ln (2) + \ln (1 + 2 x) - \ln (x - 6) \cdot 2}{2} + \ln (3) + \ln (x)$

Step 14

Multiply $2$ by $- 1$ .

$\frac{2 \ln (2) + \ln (1 + 2 x) - 2 \ln (x - 6)}{2} + \ln (3) + \ln (x)$