Practice question

Doubling time under continuous exponential growth is derived from Nt = N0e^(rt). Setting Nt to 2N0 gives 2 = e^(rT), where T is the time required to double. Taking natural logarithms yields ln 2 = rT, and rearrangement gives T = ln 2/r. Because ln 2 is approximately 0.693, a larger positive r produces a shorter doubling time. The dimensions are consistent: if r is measured per year, dividing the dimensionless logarithm by r gives years. The formula applies only when r remains constant and positive; a population with r = 0 never doubles under the model, while negative r describes decline. For discrete geometric growth the corresponding expression is T = ln 2/ln λ, not ln 2/λ, because λ is a finite multiplier. Since r = ln λ for matching time units, the two formulas are consistent. This derivation also explains why multiplying r by ln 2 cannot represent time: that product retains units of inverse time.