A Twisted Binomial Expansion!

Algebra Level 4

\[\large{f(a,\ b,\ n) = \sum_{r=0}^n {n \choose r} \log \left(a^{n-r} b^r \right) }\]

Let the above function satisfies for all positive integers \(a,b,n\). If the value of \(f(7, \ 11, \ 21)\) can be expressed as:

\[\large{\left(P_1^A \cdot P_2^B \cdot P_3^C \right) \cdot \log(D)}\]

for positive integers \(A,B,C,D\) and distinct primes \(P_1, P_2, P_3\) where \(D\) isn't any perfect \(m^{th}\) power of an integer with \(m>1\), find the value of \(P_1 + P_2 + P_3 + A+B+C+D\)?


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