Why use binary search if there's ternary search?

Solution 1:

Actually, people do use k-ary trees for arbitrary k.

This is, however, a tradeoff.

To find an element in a k-ary tree, you need around k*ln(N)/ln(k) operations (remember the change-of-base formula). The larger your k is, the more overall operations you need.

The logical extension of what you are saying is "why don't people use an N-ary tree for N data elements?". Which, of course, would be an array.

Solution 2:

A ternary search will still give you the same asymptotic complexity O(log N) search time, and adds complexity to the implementation.

The same argument can be said for why you would not want a quad search or any other higher order.