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The natural logarithm models the relationship between the number of rotations and the distance from the center of a logarithmic spiral. By taking the reciprocal of the natural logarithm, $1/\ln (x)$, the video derives the formula that matches the decreasing density of prime numbers.

Conditions: The spiral's rotation rate is modeled by $φ = \ln (r)$.; Prime density is approximated by the reciprocal $1/\ln (x)$.

The natural logarithm models the relationship between the number of rotations and the distance from the center of a logarithmic spiral. By taking the reciprocal of the natural logarithm, $1/\ln (x)$, the video derives the formula that matches the decreasing density of prime numbers.

Conditions: The spiral's rotation rate is modeled by $φ = \ln (r)$.; Prime density is approximated by the reciprocal $1/\ln (x)$.

The cumulative number of primes always increases because larger ranges contain more primes in absolute terms. However, the density decreases because the proportion of primes relative to the total number of integers searched drops as x grows.

Conditions: x increases through the displayed powers of ten.; Density is defined as (# primes)/(size).

The cumulative number of primes always increases because larger ranges contain more primes in absolute terms. However, the density decreases because the proportion of primes relative to the total number of integers searched drops as x grows.

Conditions: x increases through the displayed powers of ten.; Density is defined as (# primes)/(size).

Prime density up to x is defined as the ratio of the number of primes found to the search size x. The video expresses this operationally as # PRIMES divided by SIZE.

Conditions: x is a positive integer search bound.; Density is computed by dividing the prime count by the search size.

Prime density up to x is defined as the ratio of the number of primes found to the search size x. The video expresses this operationally as # PRIMES divided by SIZE.

Conditions: x is a positive integer search bound.; Density is computed by dividing the prime count by the search size.