Exponential growth and decay: reading the model
Separate the starting amount from the repeated growth factor, then connect each parameter to a real change per time interval.
LaTeX article Updated July 13, 2026
The initial value is the output at time zero
Substituting t = 0 makes b⁰ = 1, so A(0) = A₀. That makes the coefficient the starting amount, not the amount added each interval.
Units matter: if t counts years, the factor describes one year. A monthly factor needs t measured in months or an adjusted exponent.
Percent change becomes a multiplier
A growth rate r means the new amount is 100% + r of the old amount, so b = 1 + r. A decay rate r leaves 100% − r, so b = 1 − r.
Convert percentages to decimals before building the factor. A 6% increase uses 1.06, while a 6% decrease uses 0.94.
Exponential change is multiplicative
Equal time steps multiply by the same factor; they do not add the same amount. That is why the absolute change grows during growth and shrinks during decay.
A model is only as useful as its domain and assumptions. Real populations, prices, and substances rarely follow one unchanged factor forever.
Worked example
Common mistakes
- Using 0.04 as the growth factor.
- Treating exponential change as equal additive increases.
- Mixing the time unit and the factor's interval.
Keep these ideas
- A₀ is the value at time zero.
- Percent change becomes 1 ± r.
- Equal intervals share a multiplier.