log n is the height of a tree when the log base and the branching factor of the tree is the same.
Hash lookups are log n, because the hash table implementation is typically a tree.
Hash lookups can be close to O(1) (constant time) if you use a hash function that calculates a close-to-unique index into a table based on the key's value (provided the table isn't too full, so there are few collisions). This kind of data structure is typically used for looking up identifiers (variable and function names) in a compiler.
More information here: https://en.wikipedia.org/wiki/Hash_table