Ajitesh's function

Algebra Level 4

Consider a function \(f\) from the real numbers to the real numbers, such that \(2f(xy) = [{f(x)}]^y + [{f(y)}]^x\) for all real values \(x,y\), and \(f(1) =2\). If \(\sum_{i=1}^{100} f(i) \) can be written as \(2^a-2\), what is \(a\)?

This problem is posed by Ajitesh M.

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