Problem:
For any positive integer n, let
f(n)={log8​n,0,​ if log8​n is rational otherwise ​
What is ∑n=11997​f(n)?
Answer Choices:
A. log8​2047
B. 6
C. 355​
D. 358​
E. 585
Solution:
Since log8​n=31​(log2​n), it follows that log8​n is rational if and only if log2​n is rational. The nonzero numbers in the sum will therefore be all numbers of the form log8​n, where n is an integral power of 2. The highest power of 2 that does not exceed 1997 is 210, so the sum is:
log8​1+log8​2+log8​22+log8​23+⋯+log8​210=0+31​+32​+33​+⋯+310​=355​​
Challenge: Prove that log2​3 is irrational. Prove that, for every integer n, log2​n is rational if and only if n is an integral power of 2.