Problem:
What is the value of the expression
log2100!1+log3100!1+log4100!1+⋯+log100100!1?
Answer Choices:
A. 0.01
B. 0.1
C. 1
D. 2
E. 10
Solution:
Express each term using a base-10 logarithm, and note that the sum equals log2/log100!+log3/log100!+⋯+log100/log100!=log100!/log100!=1.
OR
Since 1/logk100! equals log100!k for all positive integers k, the expression equals log100!(2⋅3⋅⋯⋅100)=log100!100!=1.