Expand the Logarithmic Expression natural log of square root of (4x)/(x-4)

$\ln (\sqrt{\frac{4 x}{x - 4}})$

Step 1

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

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

Step 2

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

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

Step 3

Rewrite $\ln (\frac{4 x}{x - 4})$ as $\ln (4 x) - \ln (x - 4)$ .

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

Step 4

Rewrite $\ln (4 x)$ as $\ln (4) + \ln (x)$ .

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

Step 5

Simplify each term.

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

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

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

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

Step 6

Apply the distributive property.

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

Step 7

Simplify.

Cancel the common factor of $2$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

Step 8

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

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

Step 9

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

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

Step 10

Combine the numerators over the common denominator.

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

Step 11

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

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

Expand the Logarithmic Expression natural log of square root of (4x)/(x-4)

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