What calculations go on in logarithm?

Any calculations that you find easier to perform with logarithms.

What they look like?  You want/think that the logarithm can be simplified into basic addition and multiplication, right? It cant.  The logarithm has the special name because it is an entirely unique thing.  Logarithms ARE fundamental operations, just like addition and multiplication.  There is nothing special going on, no hidden arithmetic behind the scenes.

Calculators dont do exact logarithms, they do approximations.  In higher level math you will learn that functions like the logarithm can be written as a series of multiplications and additions... but only an infinite number of additions.  Calculators rely on this hidden secret formula, but obviously do not do an infinite number of computations, so they naturally return approximations.