Iteratively find and modify a certain element in a set in $\log n$ per iteration

I've looked around and can't find my answer.

I have a set of $n$ numbers. In each iteration I would like to find an arbitrary number and change it. Is it possible to perform each iteration in $\log n$ time?

There are data structures like a min-max heap which allow insertion and deletion in $\log n$ time, but that only allows deletion on the min/max values.