Similar to typical R^n, functions on a given domain can be given a vwctor space structure. Just like you can apply a change of basis to a normal vector space to make some problems simpler, you can do a similar thing for these function spaces. The Laplace transform and Fourier transform are basically doing just that. In principle, you can take any set of basis functions and define some transform to express any function in terms of your new basis. However, finsing a nice collection of basis functions and a nice way of transforming things is non-trivial which is why it is mainly those 2 transforms that get used in practice. You can do some stuff wirh power/taylor series but idk of anything that will spit out logs or other weird functions.
