Which one of the following is correct

Options

A. `1 lt log_(e) 2 lt log_(e) 3 `
B. `log_(e) 3 lt log_(e) 2 lt 1 `
C. `log_(e) 2 lt 1 lt log_(e) 3 `
D. `1 lt log_(e) 3 lt log_(e) 2 `

Answer

To solve the question regarding the logarithmic values of log base e of 2 and log base e of 3, we will analyze each logarithm step by step.

### Step 1: Understanding Logarithmic Values
We need to evaluate the logarithms:
- \( \log_e(2) \)
- \( \log_e(3) \)

### Step 2: Evaluate \( \log_e(2) \)
The expression \( \log_e(2) \) asks the question: "To what power must e (approximately 2.72) be raised to yield 2?"

Since \( e^1 = 2.72 \) and \( e^0 = 1 \), we can deduce that:
- \( 0 < \log_e(2) < 1 \)

### Step 3: Evaluate \( \log_e(3) \)
Next, we evaluate \( \log_e(3) \). This expression asks: "To what power must e be raised to yield 3?"

Since \( e^1 = 2.72 \) and \( e^2 \) is approximately \( 7.39 \), we can conclude that:
- \( 1 < \log_e(3) < 2 \)

### Step 4: Compare the Values
From the evaluations:
- \( \log_e(2) < 1 \)
- \( \log_e(3) > 1 \)

### Step 5: Conclusion
Based on the comparisons, we can summarize:
- \( \log_e(2) < 1 < \log_e(3) \)

Thus, the correct relationship among the values is:
- \( \log_e(2) < 1 < \log_e(3) \)

### Final Answer
The correct option is (C): \( \log_e(2) < 1 < \log_e(3) \).

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