Exponential Growth Calculator

For discrete growth per period, N(t) = N0 x (1 + r)^t. For continuous growth, N(t) = N0 x e^(k x t), where k = ln(1 + r). A 5% per-period discrete rate equals a continuous rate of about 4.879%.

For discrete growth, ln(2) / ln(1 + r). For continuous growth, ln(2) / k. At 5% per period that is ln(2) / ln(1.05), about 14.207 periods. The rule of 72 (72 divided by the percent rate) is a quick mental shortcut.

Use a negative rate. A 3% loss per period becomes r = -0.03, giving N(t) = N0 x 0.97^t. Starting from 100 with r = -3% over 10 periods leaves about 73.74.

Discrete applies the rate once per period; a 5% annual rate compounded yearly turns $100 into $105 after one year. Continuous assumes infinitely small intervals; the same $100 becomes about $105.13 after one year.

With initial 100, final 200, and 10 periods, the discrete rate is r = (200 / 100)^(1 / 10) - 1 = 2^0.1 - 1, about 7.177% per period. The continuous equivalent is k = ln(200 / 100) / 10 = ln(2) / 10, about 6.931%.