A line segment on the number line from 0 to 1 is marked at points corresponding to log [1, 2, 3, ..., 10], thus dividing it into 9 adjacent segments.

Question

The length of the longest such segment minus the length of the shortest such segment represents which value among these choices: log(9/2) , log(18), log (2/9), log (1.8), or log(9^2) ?

Answer 1

Note that the intervals get smaller as the input to the log function increases, so log9 to log10 would be the smallest interval and the largest interval would be log2 to log1 or just log2. So the différence requested in the problem is log2 - (log10-log9) = log2 - log(10/9) = log(18/10) = log(1.8)

Answer 2

If you didn't have a grasp for the logarithmic scale, you might start calculating distances! Bad strategy! Know what the logarithmic scale looks like, then reason as follows: The longest segment is from log(1) to log(2), and the shortest from log(9) to log(10). Writing the larger number first for each of these pairs, the calculation is therefore (log(2) - 0) - (1 - log(9)) = log(2) + log(9) - 1 = log(18) - 1 = log(1.8).

This problem tests both your spatial skill in arranging numbers by size, and also skill at manipulating logarithms.