Logarithm of the exponential

Ok, that's awkward.

E ^ Log[a]

gives onThat's what I expected, BUT,


is not evaluated at all (I expected the result) on-That's actually an MWE, which I tried was more complex).

Since I still have trouble understanding some concepts of Mathematica (such as when expressions are not rated), I thought (it seems wrong) that it was one assess or assumptions Problem, but now I think that it was not. I tried:

Provided[a [Element] Reals, Log[E^a]]// Rate
Provided[A>0Log[Exp[A>0Log[Exp[a>0Log[Exp[a>0Log[Exp[a]]]// Simplify

and other variants – which shows that I give a shot in the dark.

What is the explanation for this behavior? How do I get to on? Do I miss an assumption? I'm not really interested in any weird special cases, on is real, not complex, nor a function, nor anything else.

It seems obvious that I'm missing something (possibly very simple and obvious), but I can not figure out what.