Unless otherwise stated, logarithms are to base two. The mathematical proof of the solution by induction has two stages. It is a very good complexity to have because T n grows only slowly as n gets large. See the section on linear search. A third class of algorithms has complexity given by the following recurrence relation.
Exponential complexity grows very rapidly indeed with the size of the problem.
Just adding one to the problem size causes the complexity to roughly double. If a problem requires algorithms with this complexity and the problem size is large then its solution by computer may be possible in principle but infeasible in practice. The time or space used by an implementation of an algorithm can be measured exactly in seconds or in bytes for a particular type of computer but will be different on another computer. Thus the precise constant of proportionality for a particular computer is often not of interest. Alternatively, some operations may be chosen and the number of these executed can be used as a machine-independent measure of the time complexity.