Let's simplify the expression \(\ln \sqrt{e} + \ln \frac{1}{e}\).

### Step 1: Simplify each logarithmic term

1. **Simplify \(\ln \sqrt{e}\):**

   - The square root of \(e\) can be written as \(e^{1/2}\).
   - Using the logarithm power rule, \(\ln (e^{1/2}) = \frac{1}{2} \ln e\).
   - Since \(\ln e = 1\), we get \(\frac{1}{2} \ln e = \frac{1}{2}\).

2. **Simplify \(\ln \frac{1}{e}\):**

   - This can be rewritten using the logarithm rule for division: \(\ln \frac{1}{e} = \ln (e^{-1})\).
   - Using the power rule again, \(\ln (e^{-1}) = -\ln e\).
   - Since \(\ln e = 1\), we get \(-\ln e = -1\).

### Step 2: Combine the simplified terms

Now, add the simplified terms:

\[
\frac{1}{2} + (-1) = \frac{1}{2} - 1 = -\frac{1}{2}
\]

So, the simplified value of \(\ln \sqrt{e} + \ln \frac{1}{e}\) is \(-\frac{1}{2}\).

Would you like more details or have any questions? Here are some related questions:

1. What are the properties of logarithms that help in simplifying expressions?
2. How do you handle logarithms with different bases?
3. What is the significance of the natural logarithm in calculus?
4. How can logarithmic expressions be used to solve exponential equations?
5. How does the change of base formula work in logarithms?

**Tip:** When working with logarithms, always look for opportunities to apply rules like the power rule, product rule, and quotient rule to simplify expressions!

Simplify the logarithmic expression ln(√e) + ln(1/e) using key logarithmic properties such as the power rule and quotient rule. This step-by-step solution is ideal for high school students and covers natural logarithms and exponentiation concepts.

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