The prime number theorem states that the number \(\pi(x)\) of primes less than \(x\) is asymptotic to \(x / \ln x\). As is well known, already Chebyshev proved that \(\pi(x)\) is bounded from below and above by functions of the type \(cx/\ln x\) for certain constants \(c\), and that if \(\pi(x) \sim cx/\ln x\), then \(c = 1\). The latter result can be stated in the following form: if \(\pi(x) \sim x/\log_c x\) for some \(c > 1\), then \(c = e\). The authors ask if there is a heuristic explanation why \(c = e\), and they answer this question by proving the following result using only basic results from calculus: if \(x/\pi(x)\) is asymptotic to an increasing function, then \(\pi(x) \sim x/\ln x\). A natural candidate for such an increasing function is the upper convex hull of \(x/\pi(x)\), and in fact it can be shown using the prime number theorem that this function is asymptotic to \(\ln x\).