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Let's solve these two problems involving natural logarithms (\(\ln\)).

### Problem (a)
Evaluate:
\[
\ln(1)
\]

The natural logarithm of 1 is always 0 because:
\[
\ln(1) = 0
\]
This is true since \(e^0 = 1\).

**Answer for (a):** \(\ln(1) = 0\)

---

### Problem (b)
Evaluate:
\[
\ln\left(\frac{1}{e}\right)
\]

Using the property of logarithms, \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\), we can rewrite this as:
\[
\ln\left(\frac{1}{e}\right) = \ln(1) - \ln(e)
\]

We already know \(\ln(1) = 0\), and \(\ln(e) = 1\), so:
\[
\ln\left(\frac{1}{e}\right) = 0 - 1 = -1
\]

**Answer for (b):** \(\ln\left(\frac{1}{e}\right) = -1\)

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Would you like more details, or do you have any questions?

### Related Questions
1. What is the value of \(\ln(e^2)\) and why?
2. How does the property \(\ln(a \cdot b) = \ln(a) + \ln(b)\) help in simplifying logarithmic expressions?
3. How can the base change formula be applied to natural logarithms?
4. What are some real-world applications of natural logarithms?
5. Why is \(\ln(0)\) undefined?

### Tip
Remember that \(\ln(e^x) = x\) because of the definition of the natural logarithm as the inverse of the exponential function with base \(e\).

Evaluate without a calculator:

a. \( \ln(1) \) = ?
b. \( \ln\left(\frac{1}{e}\right) \) = ?

Learn how to evaluate natural logarithmic expressions like ln(1) and ln(1/e) without a calculator. This problem covers key logarithmic properties, including ln(1) = 0 and the property ln(a/b) = ln(a) - ln(b). Suitable for high school students in Grades 10-12.

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