Expand the Logarithmic Expression natural log of square root of (9x)/(x-9)

$\ln (\sqrt{\frac{9 x}{x - 9}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{9 x}{x - 9}}$ as ${(\frac{9 x}{x - 9})}^{\frac{1}{2}}$ .

$\ln ({(\frac{9 x}{x - 9})}^{\frac{1}{2}})$

Step 2

Expand $\ln ({(\frac{9 x}{x - 9})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \ln (\frac{9 x}{x - 9})$

Step 3

Rewrite $\ln (\frac{9 x}{x - 9})$ as $\ln (9 x) - \ln (x - 9)$ .

$\frac{1}{2} (\ln (9 x) - \ln (x - 9))$

Step 4

Rewrite $\ln (9 x)$ as $\ln (9) + \ln (x)$ .

$\frac{1}{2} (\ln (9) + \ln (x) - \ln (x - 9))$

Step 5

Simplify each term.

Rewrite $\ln (9)$ as $\ln (3^{2})$ .

$\frac{1}{2} (\ln (3^{2}) + \ln (x) - \ln (x - 9))$

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

$\frac{1}{2} (2 \ln (3) + \ln (x) - \ln (x - 9))$

Step 6

Apply the distributive property.

$\frac{1}{2} (2 \ln (3)) + \frac{1}{2} \ln (x) + \frac{1}{2} (- \ln (x - 9))$

Step 7

Simplify.

Cancel the common factor of $2$ .

Factor $2$ out of $2 \ln (3)$ .

$\frac{1}{2} (2 (\ln (3))) + \frac{1}{2} \ln (x) + \frac{1}{2} (- \ln (x - 9))$

Cancel the common factor.

$\frac{1}{2} (2 \ln (3)) + \frac{1}{2} \ln (x) + \frac{1}{2} (- \ln (x - 9))$

Rewrite the expression.

$\ln (3) + \frac{1}{2} \ln (x) + \frac{1}{2} (- \ln (x - 9))$

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

$\ln (3) + \frac{\ln (x)}{2} + \frac{1}{2} (- \ln (x - 9))$

Combine $\frac{1}{2}$ and $\ln (x - 9)$ .

$\ln (3) + \frac{\ln (x)}{2} - \frac{\ln (x - 9)}{2}$

Step 8

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

$\ln (3) \cdot \frac{2}{2} - \frac{\ln (x - 9)}{2} + \frac{\ln (x)}{2}$

Step 9

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

$\frac{\ln (3) \cdot 2}{2} - \frac{\ln (x - 9)}{2} + \frac{\ln (x)}{2}$

Step 10

Combine the numerators over the common denominator.

$\frac{\ln (3) \cdot 2 - \ln (x - 9)}{2} + \frac{\ln (x)}{2}$

Step 11

Move $2$ to the left of $\ln (3)$ .

$\frac{2 \ln (3) - \ln (x - 9)}{2} + \frac{\ln (x)}{2}$

Expand the Logarithmic Expression natural log of square root of (9x)/(x-9)

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