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[[[> The logarithms here are natural logarithms, or logarithms to base $e.$ ]]] The number of prime numbers below /x/ is asymptotic to $\frac{x}{\log~x}$ or, equivalently, to the logarithmic integral $Li(x).$ This fact is known as the "prime number theorem"; it was proved in the early 20th century by Hadamard and de la Vallee-Poussin. Informally and handwavily: "the probability that /n/ is prime is approximately EQN:1/\log(n). " (Of course this statement is nonsense if taken at face value, but for many purposes the prime numbers behave rather like random numbers selected with that density.)