1. The problem statement, all variables and given/known data Let's say I had this recurrence relation: [tex]log\left(f\left(x+2\right)\right) = log\left(f\left(x+1\right)\right) + log\left(f\left(x\right)\right)[/tex] How do I prove, then, that... [tex]f\left(x\right) = e^{c_1 L_x + c_2 F_x}[/tex] ? 2. Relevant equations There probably are some, but I don't know any. 3. The attempt at a solution I've gotten the equation to remove the logs, but I just get... [tex]f\left(x+2\right) = f\left(x+1\right)f\left(x\right)[/tex] I don't know where to go from there.
First, use the properties of the logarithm to get rid of the logarithm: [tex]log(f(x+ 2))= log(f(x+1))+ log(f(x))= log(f(x+1)f(x))[/tex] and, since log is one-to-one, f(x+2)= f(x+1)f(x). It's certainly true that the formula you gives satisfies that. Can you prove the solution is unique?