I have to apologize, I gave a slightly incorrect Mathematica code earlier, 
the actual code was:

Solve[ Exp[-2*a*x]-1+4*a*x==0,x]//N

The earlier code gave the wrong answer. Anyways:

it could be a different function, which potentially would run much longer, 
> by repetitive splitting of the interval
> (I guess that's what Mathematica is doing)  
>

There appear to be 2 main solving functions in Mathematica, "Solve" and 
"Reduce," the differences are explained in this well written answer:
https://mathematica.stackexchange.com/questions/17127/what-is-the-difference-between-reduce-and-solve

What is the result of such a Mathematica command on an example with 
> infinitely many roots given in this thread?



So, the Solve function doesn't find any it seems but it tells you to use 
the Reduce function, which I think does give all the solutions. 

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