the equation you can verify the results of table by repeatedly dividing range (from the first columnin half until it' too small to divide further the number of divisions this process requires is the number of comparisons shown in the second column repeatedly dividing the range by two is an algorithmic approach to finding the number of comparisons you might wonder if you could also find the number using simple equation of coursethere is such an equationand it' worth exploring here because it pops up from time to time in the study of data structures this formula involves logarithms (don' panic yet the numbers in table leave out some interesting data they don' answer such questions aswhat is the exact size of the maximum range that can be searched in five stepsto solve this problemwe must create similar tablebut one that starts at the beginningwith range of oneand works up from there by multiplying the range by two each time table shows how this looks for the first seven steps table powers of two step ssame as log (rrange range expressed as power of ( for our original problem with range of we can see that steps don' produce range quite big enough ( )while steps cover it handily ( thusthe steps that are shown for items in table are correctas are the steps for range of doubling the range each time creates series that' the same as raising two to poweras shown in the third column of table we can express this power as formula if represents steps (the number of times you multiply by two--that isthe power to which two is raisedand represents the rangethen the equation is