Along similar lines to Matthew Daly's answer:

Binet's formula gives an exact value for the n'th Fibonacci number where numbering starts with $F_0=0$ and $F_1=1$:

$F_n = \frac{(1+\sqrt{5})^n-(1-\sqrt{5})^n}{2^n \sqrt{5}}$

$\sqrt{5} F_n=(\frac{1+\sqrt{5}}{2})^n - (\frac{1-\sqrt{5}}{2})^n$

$=\phi^n - (\frac{-1}{\phi})^n$, where $\phi = \frac{1+\sqrt{5}}{2}$ (using the identity that $\phi-1=\frac{1}{\phi}$, which is easy to prove).

From there it's easy to show that for $n>2$, $|$log$_\phi(F_n\sqrt{5})-n|<0.5$. (Hint: as $n$ becomes large, the first of these two terms becomes very large and the second goes to zero.)

If $F_n=1$ obviously the question is unanswerable, and if $F_n=0$ it's trivial. If $F_n>1$ then $n>2$ and so we can calculate $n$ by rounding log$_\phi(F_n\sqrt{5})$ to the nearest integer.

Now we have $n$, simply apply Binet's formula in the forwards direction and we're done.