Find the exact value of the logarithmic expression ln⁡(1e) without using a calculator.If this is...

Step 1 
Given: 
Let ${\displaystyle y=\ln \left(\frac{1}{\sqrt{e}}\right)}$ 
To Find:To assess the provided expression.
Step 2 
Solution: 
${\displaystyle y=\ln \left(\frac{1}{\sqrt{e}}\right)}$ 
${\displaystyle \text{We know that }\ln \left(\frac{a}{b}\right)=\ln a-\ln b\text{ (Properties of logarithms)}}$ 
${\displaystyle \Rightarrow y=\ln 1-\ln \left(\sqrt{e}\right)}$ 
${\displaystyle \text{Since }\ln 1=0}$ 
${\displaystyle \Rightarrow y=0-\ln \left(\sqrt{e}\right)}$ 
${\displaystyle y=-{\ln \left(e\right)}^{\frac{1}{2}}}$ 
${\displaystyle \text{We know that }{\ln x}^n\text{ (Properties of logarithms)}}$ 
${\displaystyle \Rightarrow y=-\frac{1}{2}\ln e}$ 
${\displaystyle \text{Since }\ln e=1}$ 
${\displaystyle \Rightarrow y=-\frac{1}{2}\times 1}$ 
${\displaystyle y=\frac{-1}{2}}$ 
${\displaystyle \text{Hence }\ln \left(\frac{1}{\sqrt{e}}\right)=\frac{-1}{2}}$