Here's a "fun" way to figure it out: (although Galactus' seems more simple):
Notice:
f(a+b)=Aeakebk=A1f(a)f(b)
Then:
\(\displaystyle f(8) = f(4+4) = \frac{1}{A}f(4)f(4) = \frac{1}{A}f(2+2)f(2+2) \\ = \frac{1}{A} \( \frac{1}{A}f(2)f(2) + \frac{1}{A}f(2)f(2) \) = \frac{2}{A^2}
f(2)^2\)
So,
\(\displaystyle f(2)=10 \,\, \Rightarrow \,\, f(8) = \frac{2}{A^2} \( f(2) \)^2 = \frac{200}{A^2}\)
Also,
f(8)=Ae8k=A2200⇒A3e8k=200
Or for convenience sake, \(\displaystyle A^3e^{8k} = A^3\(e^{2k}\)^4 =
Ae2k^3e^{2k}=200.\)
Therefore we have a system of equations:
1. Ae2k=10
2. \(\displaystyle \,\, \(Ae^{2k}\)^3e^{2k}=200\)
Substituting (1) into (2) gives an easy method for solving.
edit: For added fun (yeah, right): See if you can find f(8) without knowing what k is.