I'm reading a paper on Randomized Rumor Spreading and I'm confused about part 2.1. startup phase (page 4).
We try to spread a message ("rumor") using a push-scheme, where each informed player calls a random player in each round and sends the message. At the beginning of the first round only one player holds the rumor. If we execute c rounds then probability that this player has at least once called an uninformed player (i.e., didn't call itself) is $1-n^{-c}$, where n is the number of players in the network. So the number of informed players doubles each round with high probability, when the number of informed nodes is small.
The next step is confusing to me. How does it follow from the fact that it takes at most c rounds to double the number of informed nodes, that it takes at most $O(\ln(\ln n))$ rounds to achieve $(\ln n)^4$ players?