Question

The value of the expression: \[ \displaystyle \sum_{i=2}^{100} \frac{1}{\log_i(100!)} \] is:

• A. 0.01
• B. 0.1
• C. 1
• D. 10
• E. 100

Hint:

When you see $\tfrac{1}{\log_b a}$, always convert it to $\log_a b$. This often turns sums into telescoping or into neat factorial/log simplifications.

Answer 1

The Correct Option is C

Step 1: Simplify reciprocal log.
\[ \frac{1}{\log_i(100!)} = \log_{100!}(i). \] Step 2: Rewrite the sum.
\[ \sum_{i=2}^{100}\frac{1}{\log_i(100!)}=\sum_{i=2}^{100}\log_{100!}(i). \] Step 3: Combine logs.
Using $\log_a b+\log_a c=\log_aB(C)$: \[ \sum_{i=2}^{100}\log_{100!}(i)=\log_{100!}(2 3 4s 100). \] Step 4: Recognize factorial.
$2 3 4s 100=100!$. So \[ \log_{100!}(100!)=1. \] \[ \boxed{1} \]