In a nutshell
Exponential search first finds a small window where the target must live, then binary-searches it. Starting at index 1 it keeps doubling the bound — 1, 2, 4, 8, 16… — until that position's value exceeds the target. The target then lies between the last two bounds, a range it finishes off with ordinary binary search. Because it homes in before searching, it's ideal for unbounded or very large sorted lists, finding a target near position i in about log i steps.