Discrete and continuous models
Discrete growth adds a fixed percentage once per period: x(t) = x₀(1 + r)ᵗ. Continuous growth compounds without a break: x(t) = x₀eᵏᵗ. The two describe the same growth when k = ln(1 + r).
To solve for the other quantities: x₀ = x(t) ÷ (1 + r)ᵗ, r = (x(t)/x₀)^(1/t) − 1, t = ln(x(t)/x₀) ÷ ln(1 + r). In the continuous model, k = ln(x(t)/x₀)/t and t = ln(x(t)/x₀)/k.
Worked example
1,000 growing at 5% per period for 10 periods: x(10) = 1,000 × 1.05¹⁰ = 1,628.89. It doubles every ln 2 ÷ ln 1.05 = 14.21 periods, and the equivalent continuous rate is k = ln 1.05 = 4.879%.
At a continuous 5%, the same 10 periods give 1,000 × e^0.5 = 1,648.72.
Doubling time and half-life
Doubling time is ln 2 divided by the per-period log growth rate. When the rate is negative, the same formula gives the half-life. For radioactive decay stated in half-lives, use the half-life calculator.