To simplify the expression \(\ln\left(\frac{1}{e}\right) - \frac{1}{2}\ln(e)\), let's work through each part step by step.

### Step 1: Simplify \(\ln\left(\frac{1}{e}\right)\)
Recall that \(\ln\left(\frac{1}{e}\right)\) is the natural logarithm of a fraction. Using logarithmic rules:

\[
\ln\left(\frac{1}{e}\right) = \ln(1) - \ln(e)
\]

Since \(\ln(1) = 0\) and \(\ln(e) = 1\), we have:

\[
\ln\left(\frac{1}{e}\right) = 0 - 1 = -1
\]

### Step 2: Simplify \(\frac{1}{2}\ln(e)\)
Since \(\ln(e) = 1\), we have:

\[
\frac{1}{2}\ln(e) = \frac{1}{2} \times 1 = \frac{1}{2}
\]

### Step 3: Combine the results
Now, we can combine the simplified terms:

\[
-1 - \frac{1}{2} = -\frac{3}{2}
\]

Thus, the simplified expression is:

\[
\boxed{-\frac{3}{2}}
\]

Would you like further details or have any questions?

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Here are 5 questions to extend this concept:
1. What are the logarithmic properties that allow simplifications like these?
2. How would the expression change if it was \(\ln\left(\frac{1}{x}\right) - \frac{1}{2}\ln(x)\)?
3. What is the logarithmic identity for \(\ln(a \cdot b)\)?
4. How does \(\ln\) behave when dealing with exponents, such as \(\ln(x^n)\)?
5. How do natural logarithms differ from logarithms with other bases?

**Tip:** Always remember the key logarithmic identities such as \(\ln(ab) = \ln(a) + \ln(b)\) and \(\ln(a^b) = b \cdot \ln(a)\) to simplify expressions!

Simplify the logarithmic expression ln(1/e) - 1/2ln(e) using key logarithmic identities such as ln(a/b) = ln(a) - ln(b) and ln(e) = 1. A step-by-step solution is provided, suitable for high school students in Grades 9-12.

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