For a acid-base titration, why does the pH change slowly at first and then have a rapid increase as it nears the equivalence point?

The pH increases slowly at first because the pH scale is logarithmic, which means that a pH of 1 will have 10 times the hydronium ion concentration than a pH of 2. Thus, as the hydronium ion is initally removed, it takes a lot of base to change its concentration by a factor of 10, but as more and more hydronium ion is removed, less base is required to change its concentration by a factor of 10. Near the equivalence point, a change of a factor of 10 occurs very quickly, which is why the graph is extremely steep at this point. As the hydronium ion concentration becomes very low, it will again take a lot of base to increase the hydroxide ion concentration by 10 fold to change the pH significantly.