Why do logarithms appear in the prime number theorem?

The appearance of logarithms in the prime number theorem is considered an extraordinary connection between the discrete nature of prime numbers and the continuous nature of logarithms.1

The theorem states that for large values of $x$, the number of primes less than or equal to $x$, denoted by $\pi (x)$, is approximately $x/\ln (x)$. This relationship implies that the density of primes near a number $n$ is roughly $1/\ln (n)$.21

While early mathematicians like Carl Friedrich Gauss observed this pattern through the empirical study of prime tables, the theoretical link was solidified through the work of Bernhard Riemann. Riemann connected the distribution of primes to the Riemann zeta function, a function of a complex variable. By analyzing this function, mathematicians were able to bridge the gap between integer properties and analytic functions, ultimately proving in 1896 that the ratio of $\pi (x)$ to $x/\ln (x)$ approaches 1 as $x$ increases toward infinity.1342