n^th Fibonacci number requires O(n) bits. So, saying the solution is O(log(n)) is kinda not true. I mean there cannot be a o(n) algorithm for computing decimal representation of n^th fibonacci number.
As always, it depends on your computational model. People are awful at specifying a model, in other words it's unclear what's even being stated.
You can compute the nth fibonacci number in O(lg n) additions and multiplies. That indeed doesn't turn out to be a good model for how long it takes on an actual computer though.
And even if the model is sufficiently specified, the constant factors and lower terms hidden in the O-notationen can be deciding in practice, and are yet usually left out.