Question

The value of \( \sum_{a,b,c} \frac{1}{1 + \log_a bc} \) is:

• A. 0
• B. 1
• C. 2
• D. 3

Hint:

When dealing with symmetric logarithmic sums, try expressing each term in terms of natural logs to reveal patterns or identities that simplify the sum.

Answer 1

The Correct Option is B

Step 1: Expand the sum.
Assume \( a, b, c \) are distinct positive numbers. The sum is: \[ \frac{1}{1 + \log_a bc} + \frac{1}{1 + \log_b ca} + \frac{1}{1 + \log_c ab} \] Step 2: Use identity.
Use: \( \log_a bc = \log_a b + \log_a c \) Also, recall the identity: \[ \frac{1}{1 + \log_a bc} = \frac{1}{1 + \frac{\log bc}{\log a}} = \frac{\log a}{\log a + \log bc} \] So the full expression becomes: \[ \frac{\log a}{\log a + \log b + \log c} + \frac{\log b}{\log a + \log b + \log c} + \frac{\log c}{\log a + \log b + \log c} = 1 \]