Question

Evaluate \( \log_{e^{2000}} e^{2021} \cdot \log_{e^{2021}} e^{2022} \cdot \log_{e^{2022}} e^{2023} \cdot \log_{e^{2023}} e^{2024} \)

• A. 2020
• B. 0
• C. 2024
• D. 1

Hint:

Convert each log to fraction form and observe telescoping patterns for simplification.

Answer 1

The Correct Option is B

Using the identity \( \log_a b = \frac{1}{\log_b a} \), we get:
\[ \log_{e^{2000}} e^{2021} = \frac{2021}{2000},\quad \log_{e^{2021}} e^{2022} = \frac{2022}{2021}, \quad \log_{e^{2022}} e^{2023} = \frac{2023}{2022}, \quad \log_{e^{2023}} e^{2024} = \frac{2024}{2023} \]
Multiplying all: \[ \frac{2021}{2000} \cdot \frac{2022}{2021} \cdot \frac{2023}{2022} \cdot \frac{2024}{2023} = \frac{2024}{2000} \]
Now, take \( \log_{\frac{2024}{2000}} 1 = 0 \)