Question

Which is the correct order for a given number \( \alpha \) in increasing order?

• A. \( \log_2 \alpha, \log_3 \alpha, \log_e \alpha, \log_{10} \alpha \)
• B. \( \log_{10} \alpha, \log_3 \alpha, \log_e \alpha, \log_2 \alpha \)
• C. \( \log_{10} \alpha, \log_2 \alpha, \log_e \alpha, \log_3 \alpha \)
• D. None of these

Hint:

For increasing order of logarithmic functions, the base of the logarithm determines the rate of growth, with smaller bases resulting in smaller values for \( \alpha>1 \).

Answer 1

The Correct Option is B

Step 1: Understand logarithms.
Logarithms with different bases behave differently. We are given four different logarithmic functions: \( \log_2 \alpha \), \( \log_3 \alpha \), \( \log_e \alpha \), and \( \log_{10} \alpha \).

Step 2: Compare the values for different logarithmic bases.
To determine the correct order, recall that: - \( \log_2 \alpha \) increases faster than \( \log_3 \alpha \), which increases faster than \( \log_e \alpha \), which in turn increases faster than \( \log_{10} \alpha \) when \( \alpha>1 \). - Therefore, \( \log_{10} \alpha<\log_3 \alpha<\log_e \alpha<\log_2 \alpha \) for \( \alpha>1 \).

Step 3: Conclusion.
Thus, the correct increasing order is \( \log_{10} \alpha, \log_3 \alpha, \log_e \alpha, \log_2 \alpha \), corresponding to option (B).