Exponential Growth Calculator

Exponential growth is growth whose rate is proportional to the current value, which produces a J-shaped curve that climbs slowly at first and then bends sharply upward. The formula x(t) = x₀ × (1 + r)^t multiplies the starting value x₀ by a constant growth factor (1 + r) every time period t. A population of 1,000 growing 10% a year reaches 1,100 after one year and 2,594 after ten years, because each period's growth builds on the last.

The exponential growth formula has two equivalent forms. The discrete form is x(t) = x₀ × (1 + r)^t, used when growth compounds once per period. The continuous form is x(t) = x₀ × e^(kt), where e is Euler's number, approximately 2.71828, and k is the continuous growth rate. Both forms return the value x(t) at time t from the initial value x₀. A 5% discrete rate corresponds to a continuous rate of k = ln(1.05), approximately 0.04879.

Linear growth adds a fixed amount every period, while exponential growth multiplies by a fixed factor every period. Saving $100 per period for 10 periods reaches $1,000 under linear growth, but the same $100 growing 5% per period compounds to $162.89 by period 10. Extend the comparison to 50 periods and linear growth reaches $5,000 while exponential growth reaches $1,146.74, overtaking linear growth by a wide margin at long horizons.

Doubling time is the number of periods an exponentially growing quantity takes to double in size, using the formula t₂ = ln(2) / ln(1 + r), approximately 0.693 / r for small rates. At a 7% annual growth rate, doubling time works out to ln(2) / ln(1.07), approximately 10.24 years. At a faster 10% rate, the same quantity doubles in only 7.27 years, showing how sensitive doubling time is to small changes in the growth rate.

Half-life is the time a quantity undergoing exponential decay takes to fall to half its starting value, following the same structure as doubling time: t½ = ln(2) / |k|, where k is the decay constant. Caffeine in the human body has a half-life of about 6 hours, so a 95 mg dose falls to 47.5 mg after 6 hours and 23.75 mg after 12 hours. Carbon-14 has a half-life of 5,730 years, letting radiocarbon dating estimate ages up to roughly 50,000 years.

Divide the percentage by 100 to get the decimal growth rate, then add 1 to get the growth factor used in the formula. A 5% growth rate becomes 0.05 as a decimal and 1.05 as a growth factor. A minus 3% rate, which represents decay, becomes minus 0.03 as a decimal and 0.97 as a growth factor. This conversion has to happen before the rate is raised to the power t, because the formula only works with the decimal form.

Calculate exponential growth by multiplying the initial value x₀ by the growth factor (1 + r) raised to the power t, the number of time periods. For an initial value of 1,000, a growth rate of 5% and 10 time periods, the calculation is 1,000 × 1.05¹⁰, which equals 1,628.89. Converting the rate to a decimal first matters here. Entering 5 instead of 0.05 would give a wildly different, incorrect result.

Calculate exponential decay the same way as growth, but multiply by (1 − r) raised to the power t instead of (1 + r), or use e^(−kt) for the continuous form. A 95 mg dose of caffeine with a 6-hour half-life leaves 95 × 0.5^(t / 6) mg in the body at time t. After 6 hours that is 47.5 mg, and after 12 hours it is 23.75 mg. The decay rate r and decay constant k both describe the same shrinking process.

Discrete exponential growth compounds once per period using the formula (1 + r)^t, while continuous exponential growth compounds infinitely often using e^(kt). The two produce nearly identical results at everyday rates. A 5% nominal rate yields a growth factor of 1.05000 discretely and 1.05127 continuously over one year, a difference of about 0.13%. That gap between discrete and continuous compounding grows larger as the rate increases, which matters more for high-rate financial and biological models.

Logistic growth is growth that follows an S-shaped curve and slows down as it approaches a fixed carrying capacity K, unlike exponential growth, which has no upper bound and keeps multiplying forever. The formula P(t) = K / (1 + ((K − P₀) / P₀) × e^(−rt)) adds a braking term that plain exponential growth lacks. Real populations, bacterial colonies and viral epidemics typically follow exponential growth early on, then transition to logistic growth once resources or susceptible hosts run low.

Yes, time can be negative in exponential growth. Using a negative value for t projects the same formula backwards to find a past value instead of a future one. A population of 17,103 people growing at 5% per year was 17,103 × 1.05^(−11), or 10,000 people, eleven years earlier. Negative time only works cleanly when the growth rate has stayed constant over that entire span, so long backward projections are rough estimates, not certainties.

Real-world examples of exponential growth include compound interest on savings and investments, bacterial reproduction through binary fission, and viral outbreaks during their early spreading phase. Moore's law, which describes transistor counts on computer chips roughly doubling every two years, is another well-known case. Exponential decay, the negative-rate version of the same formula, shows up in radioactive decay, radiocarbon dating and drug metabolism, such as caffeine leaving the bloodstream at a steady percentage rate.