Recurrence Relation

[Char. Limit]

1. The problem statement, all variables and given/known data
Let's say I had this recurrence relation:

log\left(f\left(x+2\right)\right) = log\left(f\left(x+1\right)\right) + log\left(f\left(x\right)\right)

How do I prove, then, that...

f\left(x\right) = e^{c_1 L_x + c_2 F_x}

?

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...

f\left(x+2\right) = f\left(x+1\right)f\left(x\right)

I don't know where to go from there.

[HallsofIvy]

First, use the properties of the logarithm to get rid of the logarithm:
log(f(x+ 2))= log(f(x+1))+ log(f(x))= log(f(x+1)f(x))
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?