Let f be a real valued function satisfying
`2f(xy) = {f(x)}^(y) + {f(y)}^(x), AA x, y in R` and f(1) = 2, then find `underset(K = 1)overset(2008)sum f(K)`

Explanation

To solve the problem, we need to find the function \( f(x) \) that satisfies the given functional equation and then compute the sum \( \sum_{k=1}^{2008} f(k) \).

### Step-by-Step Solution:

1. **Understanding the Functional Equation:**
We start with the equation:
\[
2f(xy) = f(x)^y + f(y)^x \quad \forall x, y \in \mathbb{R}
...