Exponential functional equations?

Algebra Level 4

\[\large f(2^x)=2^{f(x)}\]

Let \(f\) satisfy the equation above with \(f(0) = 1\). If \(f(2048) \) can be expressed as \(a^b\), where \(a\) and \(b\) are positive integers with \(a\) prime, find \(a+b\).

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