Expand the Logarithmic Expression natural log of square root of (6x)/(x-6)

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

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{6 x}{x - 6}}$ as ${(\frac{6 x}{x - 6})}^{\frac{1}{2}}$ .

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

Step 2

Expand $\ln ({(\frac{6 x}{x - 6})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

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

Step 3

Rewrite $\ln (\frac{6 x}{x - 6})$ as $\ln (6 x) - \ln (x - 6)$ .

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

Step 4

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

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

Step 5

Simplify each term.

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

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

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

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

Step 6

Apply the distributive property.

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

Step 7

Simplify.

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

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

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

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

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

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

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

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

Step 8

Combine the numerators over the common denominator.

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