I came across the formula for the $n$th Fibonacci number:
$$\frac{\Phi^n-(-\Phi)^n}{\sqrt5} = x,$$
where $x$ is the $n$th Fibonacci number.
This formula works one way around, but I cannot seem to get it to flip around. How would I go about isolating $n$?
A "practical" solution could be the following:
For large enough $n$, you can write the $n^{th}$ Fibonacci number as $$y_n = \frac{1}{\sqrt{5}} \left( \frac{1+\sqrt{5}}{2} \right)^{n+1}$$ where index $n$ starts from 0.
Now you can solve for $n$ from $y_n = x$ as:
$$ n = \frac{\log(x \sqrt{5})}{\log( (1+\sqrt{5})/2)} - 1 = \frac{\log(x) + 0.8047}{0.4812}-1 = 2.0781\log(x)+0.6723$$.
The $\log$ is the natural logarithm. You can round the above expression to the nearest integer.
For example, with $x=233$, I get $n=12.0001$, which I can round off to 12. Indeed, with the 0 based indexing, $y_{n+1} = y_{13} = 233$.