Lengths of [number] representations

(a) n, (b) Floor[Log[2, n] + 1], (c) Tr[FixedPointList[Max[0, Ceiling[Log[2, #]]] &, n + 2]] - n - 3, (d) 2 Ceiling[Log[3, 2n + 1]], (e) Floor[Log[GoldenRatio, √5 (n + 1/2)]]. Large n approximations: (a) n, (b) Log[2, n], (c) Log[2, n] + Log[2, Log[2,n ]] + …, (d) 2 Log[3, n], (e) Log[GoldenRatio, n].

Shown on a logarithmic scale, representations (b) through (e) (given here for numbers 1 through 500) all grow roughly linearly: