What is the largest integer less than \(\log_2 \frac{2}{1} + \log_2 \frac{3}{2} + \cdots + \log_2 \frac{2009}{2008} + \log_2 \frac{2010}{2009}\)?
+33652 log(b/a) = log(b) - log(a)
so
log2(2/1) = log2(2) - log2(1)
log2(3/2)=log2(3) - log2(2)
...
log2(2010/2009) = log2(2010) - log2(2009)
Adding up all these, resuts in most of them cancelling out, and we are left with
log2(2010) - log2(1)
Now 210 = 1024 and 211 = 2048, so I'll leave you to decide the final steps.
