Expected Rounds to Double a Kelly Bankroll

A gambler bets the Kelly fraction on an even-money coin with win probability $p=0.6$ every round, so log-wealth is a random walk with positive drift. Let $G$ be the per-round expected log-growth (the maximal Kelly growth rate). Using an optional-stopping argument on a suitable martingale, estimate the expected number of rounds until wealth first doubles. You may ignore overshoot past the doubling level.