Problem:
For any positive integer n, let
f(n)={log8n,0, if log8n is rational otherwise
What is ∑n=11997f(n)?
Answer Choices:
A. log82047
B. 6
C. 355
D. 358
E. 585
Solution:
Since log8n=31(log2n), it follows that log8n is rational if and only if log2n is rational. The nonzero numbers in the sum will therefore be all numbers of the form log8n, where n is an integral power of 2. The highest power of 2 that does not exceed 1997 is 210, so the sum is:
log81+log82+log822+log823+⋯+log8210=0+31+32+33+⋯+310=355
Challenge: Prove that log23 is irrational. Prove that, for every integer n, log2n is rational if and only if n is an integral power of 2.