Question: f(n) = log n^2; g(n) = log n + 5 => f(n) = Θ (g(n))

I think we can prove this for omega but how can we prove it for Big oh ?

because if we simplify it to logn + logn <= logn +5 => logn<=5 is not true !

Answer: 
You might be looking for a [dictionary](https://en.wikipedia.org/wiki/Dictionary_(data%20structure)). Common implementations are [hash tables](https://en.wikipedia.org/wiki/Hash_table) and [binary search trees](https://en.wikipedia.org/wiki/Binary_search_tree).
Another common name for such a "map" data structure is *Map*. That's what it is called in [C++](https://cplusplus.com/reference/map/map/), [Java](https://docs.oracle.com/en/java/javase/19/docs/api/java.base/java/util/Map.html), [ECMAScript](https://tc39.es/ecma262/#sec-map-objects), [PHP](https://php.net/manual/en/class.ds-map.php), [Go](https://go.dev/ref/spec#Map_types), [Matlab](https://mathworks.com/help/matlab/matlab_prog/overview-of-the-map-data-structure.html), [Rust](https://doc.rust-lang.org/std/collections/index.html), [Scala](https://docs.scala-lang.org/overviews/collections-2.13/maps.html), [Clojure](https://clojure.org/reference/data_structures#Maps), [Elixir](https://hexdocs.pm/elixir/main/Map.html), [OCaml](https://v2.ocaml.org/api/Map.html), [F#](https://fsharp.github.io/fsharp-core-docs/reference/fsharp-collections-fsharpmap-2.html), [Racket](https://docs.racket-lang.org/rebellion/Sorted_Maps.html), [Kotlin](https://kotlinlang.org/docs/collections-overview.html#map), [Dart](https://dart.dev/guides/language/language-tour#maps), [Ada](http://ada-auth.org/standards/rm12_w_tc1/html/RM-A-18-4.html), and [Haskell](https://hackage.haskell.org/package/containers-0.4.0.0/docs/Data-Map.html), to name just a few.

The name *Map* is also [used in Computer Science](https://tildesites.bowdoin.edu/%7Eltoma/teaching/cs210/fall09/Slides/Maps.pdf).

Yet another common name is [*Associative Array*](https://wikipedia.org/wiki/Associative_array).

A slightly less common name is *Table*. That's what it is called in [*Lua*](https://lua.org/manual/5.4/manual.html#2.1). The term *Table* also appears in terms like [*Hash Table*](https://wikipedia.org/wiki/Hash_table).