Every semigroup \(T\) on a Banach space \(X\) can be used to define elements \(u\in X\) of exponential type relative to \(T\) by requiring that \(u= T(s)v\) for some \(s>0\) and \(v\in X\). Let \(X\) and \(Y\) be Banach spaces in which the exponential type is characterized by the semigroups \(T\) and \(S\), respectively, and \(L: X\to Y\) be Fredholm. The author investigates the problem when every solution to \(Lu= f\) is of exponential type relative to \(X\) provided \(f\) is of exponential type to \(S\). Applications to certain classes of differential operators are given.