Question

The value of loge \(e{\sqrt{e}}\) is

• A. \(\frac{1}{2}\)
• B. \(\frac{2}{2}\)
• C. \(\frac{3}{2}\)
• D. \(\frac{4}{2}\)

Answer 1

The Correct Option is C

To solve this problem, we need to evaluate the expression \( \log_e (e^{\sqrt{e}}) \).

1. Applying Logarithmic Identity:
We use the identity: \( \log_b (b^x) = x \). In this case, the base \( b = e \), so the identity becomes:

\[ \log_e (e^{\sqrt{e}}) = \sqrt{e} \]

2. Estimating the Value of \( \sqrt{e} \):
We know that the value of \( e \approx 2.718 \). Therefore:

\[ \sqrt{e} \approx \sqrt{2.718} \approx 1.65 \]

3. Evaluating the Options:
Now let's compare the approximate value of \( \sqrt{e} \approx 1.65 \) with the given options:

• (1) \( \frac{1}{2} = 0.5 \)
• (2) \( \frac{2}{2} = 1 \)
• (3) \( \frac{3}{2} = 1.5 \)
• (4) \( \frac{4}{2} = 2 \)

The closest and most accurate approximation is \( \frac{3}{2} \).

Final Answer:
The correct answer is (C) \( \frac{3}{2} \).