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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 $1/\ln (x)$.