To find the exact value of the expression [tex]\(\ln \frac{1}{\sqrt{e}}\)[/tex], we will step through the properties of logarithms. Here is the detailed solution:
1. Expression Rewrite using Logarithm Properties:
The given expression is [tex]\(\ln \left(\frac{1}{\sqrt{e}}\right)\)[/tex]. Using the property of logarithms that states [tex]\(\ln \left(\frac{a}{b}\right) = \ln a - \ln b\)[/tex], we can rewrite the logarithm of a fraction:
[tex]\[
\ln \left(\frac{1}{\sqrt{e}}\right) = \ln(1) - \ln(\sqrt{e})
\][/tex]
2. Evaluate [tex]\(\ln(1)\)[/tex]:
It is known that the natural logarithm of 1 is always 0:
[tex]\[
\ln(1) = 0
\][/tex]
3. Simplify [tex]\(\ln(\sqrt{e})\)[/tex]:
Next, we need to simplify [tex]\(\ln(\sqrt{e})\)[/tex]. Recall that the square root of [tex]\(e\)[/tex] can be written as [tex]\(e^{1/2}\)[/tex]. Using the logarithmic property that states [tex]\(\ln(a^b) = b \ln(a)\)[/tex]:
[tex]\[
\ln(\sqrt{e}) = \ln(e^{1/2}) = \frac{1}{2} \ln(e)
\][/tex]
4. Evaluate [tex]\(\ln(e)\)[/tex]:
We know that the natural logarithm of [tex]\(e\)[/tex] is 1:
[tex]\[
\ln(e) = 1
\][/tex]
Therefore,
[tex]\[
\ln(\sqrt{e}) = \frac{1}{2} \cdot 1 = \frac{1}{2}
\][/tex]
5. Combine the Results:
Substitute the values we found back into the expression:
[tex]\[
\ln \left(\frac{1}{\sqrt{e}}\right) = \ln(1) - \ln(\sqrt{e}) = 0 - \frac{1}{2} = -\frac{1}{2}
\][/tex]
Thus, the exact value of the logarithmic expression [tex]\(\ln \left(\frac{1}{\sqrt{e}}\right)\)[/tex] is:
[tex]\[
-\frac{1}{2}
\][/tex]
