Expand the Logarithmic Expression natural log of square root of ex

$\ln (\sqrt{e x})$

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

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

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

$\frac{1}{2} \ln (e x)$

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

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

The natural logarithm of $e$ is $1$ .

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

Apply the distributive property.

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

Multiply $\frac{1}{2}$ by $1$ .

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

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

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