Expand the Logarithmic Expression natural log of square root of x^2(X+2)

$\ln (\sqrt{x^{2} (X + 2)})$

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

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

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

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

Rewrite $\ln (x^{2} (X + 2))$ as $\ln (x^{2}) + \ln (X + 2)$ .

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

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

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

Apply the distributive property.

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

Cancel the common factor of $2$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

$\ln (x) + \frac{\ln (X + 2)}{2}$

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

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

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

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

Combine the numerators over the common denominator.

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

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

$\frac{2 \ln (x) + \ln (X + 2)}{2}$