1998 AHSME Problems/Problem 22
Contents
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 1
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)
$$ (Error compiling LaTeX. Unknown error_msg)\frac{1}{\log_k 100!} = \frac{\log k}{\log 100!}
x=y
\frac{1}{x}=\frac{1}{y}
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)
Solution 2
Since 
,
![\[\frac{1}{\log_{k}100!}=\frac{\log_{k}k}{\log_{k}100!}=\log_{100!} k\]](http://latex.artofproblemsolving.com/4/6/c/46cf706afa1d9d44629a51dc59bbc9945ee6840d.png)
We add:
![\[\log_{100!} 1 +\log_{100!} 2 +\log_{100!} 3 +\cdots + \log_{100!} 100=\log_{100!}100!=1 \Rightarrow \mathrm{(C)}\]](http://latex.artofproblemsolving.com/5/c/f/5cfa5d82f6aedc41a17f9f60149ce86adcf21b83.png)
See also
| 1998 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 21 |
Followed by Problem 23 | |
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| All AHSME Problems and Solutions | ||
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