Expand the Logarithmic Expression natural log of 1/( square root of e)

$\ln (\frac{1}{\sqrt{e}})$

Rewrite $\ln (\frac{1}{\sqrt{e}})$ as $\ln (1) - \ln (\sqrt{e})$ .

$\ln (1) - \ln (\sqrt{e})$

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

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

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

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

The natural logarithm of $1$ is $0$ .

$0 - (\frac{1}{2} \ln (e))$

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

$0 - (\frac{1}{2} \cdot 1)$

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

$0 - \frac{1}{2}$

Subtract $\frac{1}{2}$ from $0$ .

$- \frac{1}{2}$