Succinctly:

  lim_(n→∞) bits_needed(3ⁿ) / n
    = lim_(n→∞) ⌈log2(3ⁿ)⌉ / n
    = lim_(n→∞) ⌈n log2(3)⌉ / n
    = log2(3)
    = log(3) / log(2)

where bits_needed(m) is the minimum number of bits needed to encode m distinct values.