In[1]:= FindInstance[Product[Exp[a[i]], {i, 1, n}] == Exp[Sum[a[i], {i, 1, n}]], n]
Out[1]= {{n -> 0}}

That's for sure. Are there more instances?

FindInstance[Product[Exp[a[i]], {i, 1, n}] == Exp[Sum[a[i], {i, 1, n}]], n, Integers, 2]
FindInstance::exvar: The system contains a nonconstant expression i independent of variables {n}. >>

(* input bounced back *)

One of the typical problems in that area. Others are the existence of intermediate expressions (convergence of products and sums). Even if convergence will be guaranteed, $Mathematica$ distrusts:

In[61]:= FullSimplify[Sum[a[i], i] - Log[Product[Exp[a[i]], i]], 
 Assumptions -> Exists[{x, y}, Product[Exp[a[i]], i] == x && Sum[a[i], i] == y && x == Exp[y]]]

Out[61]= -Log[Product[E^a[i], i] + Sum[a[i], i]

So far for $n\rightarrow\infty$.Seemingly the nearest thing to do is for finite $n$

In[77]:= Times @@ (Exp /@ v_?VectorQ) == Exp[Plus @@ v_?VectorQ]
Out[77]= True

If one creates a little quirk, $Mathematica$ finds about it

In[90]:= Times @@ (Exp /@ (v_?VectorQ)) == Exp[Plus @@ (v_List)]
Out[90]= E^(VectorQ + v_) == E^(v + _List)

For what it's worth, I suspect this is not a simple task. Symbolic sums and products that do not evaluate to other forms are just not easy to work with. Often people come up with creative approaches though, and I am curious to see if any such shows up in response to this.