1998 AHSME Problems/Problem 22
Problem
What is the value of the expression
![\[\frac{1}{\log_2 100!} + \frac{1}{\log_3 100!} + \frac{1}{\log_4 100!} + \cdots + \frac{1}{\log_{100} 100!}?\]](http://latex.artofproblemsolving.com/e/6/3/e63e3b9ec89bd66181c2082ca2c3f5d3837f4853.png)
Solution
By the change-of-base formula,
![\[\log_{k} 100! = \frac{\log 100!}{\log k}\]](http://latex.artofproblemsolving.com/a/7/d/a7d308108bdf05a602b6a13f026286818a8ea2e5.png)
Thus (you might recognize this identity directly)
![\[\frac{1}{\log_k 100!} = \frac{\log k}{\log 100!}\]](http://latex.artofproblemsolving.com/1/8/0/1800fe46a3d13afe1c569797b2b1d3cc3889b266.png)
Thus the sum is
![\[\left(\frac{1}{\log 100!}\right)(\log 1 + \log 2 + \cdots + \log 100) = \frac{1}{\log 100!} \cdot \log 100! = 1 \Rightarrow \mathrm{(C)}\]](http://latex.artofproblemsolving.com/4/4/6/4466915f14bf941652009fb4bc96fb00f46c8d51.png)
See also
| 1998 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 21 |
Followed by Problem 23 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
