Practice question

For continuous exponential growth, doubling time T2 satisfies 2N0 = N0e^(rT2). Cancelling N0 and taking natural logarithms gives ln 2 = rT2, so T2 = ln 2/r. With r = 0.1 per stated time unit, T2 = 0.693/0.1 = 6.93, approximately 7 time units. Direct substitution confirms the result: e^(0.1 × 6.93) ≈ e^0.693 ≈ 2. A value of 2.1 would yield e^0.21 ≈ 1.23, far short of doubling. The units of doubling time are the reciprocal of the units used for r; for example, r per year produces a doubling time in years. This calculation assumes a constant positive r and no density-dependent slowing. Geometric growth uses a related expression, T2 = ln 2/ln λ, because r = ln λ for equivalent time intervals. Thus the keyed value is inconsistent with the standard exponential-growth equation; the scientifically supported numerical choice is 7. The discrepancy is numerical, not a matter of alternative ecological terminology or convention.