When the population size is very far below the carrying capacity.

Proof 1: The logistic equation in discrete time is:

n[t+1] = n[t] + r n[t] (1-n[t]/K)

If n[t] is very small relative to K, then (1-n[t]/K) will be nearly 1. Then the equation becomes:

n[t+1] ~ n[t] + r n[t],

which is the equation describing exponential growth in discrete time (n[t+1] = R n[t], where R = 1+r).

Proof 2: The logistic equation in continuous time is:

dn/dt = r n (1-n/K)

If n is very small relative to K, then (1-n/K) will be nearly 1. Then the equation becomes:

dn/dt = r n

which is the equation describing exponential growth in continuous time (dn/dt = r n).