$lcm(1, ,n)$ Exceeds $n (n+1)$ from $n = 3$

The least common multiple of the first $n$ integers grows like $e^n$, a fact equivalent to the prime number theorem. Comparisons of that quantity against polynomial or logarithmic expressions do not need the analytic result, and this note isolates how little is required. Two consecutive integers are coprime and both divide $\\operatorname{lcm}(1,\\dots,n)$, so their product divides it too, giving the elementary bound $\\operatorname{lcm}(1,\\dots,n) \\ge n(n-1)$. Since $\\log(n+1) < n-1$ from $n = 3$, the inequality $\\operatorname{lcm}(1,\\dots,n) > n \\log(n+1)$ holds at every $n \\ge 3$, and the set of such $n$ has density one. A quadratic lower bound already outruns the logarithmic comparison, several orders of magnitude short of the truth.