Exponential Growth Calculator

What exponential growth means

Exponential growth means a quantity is multiplied by the same factor in every period, instead of having the same amount added. Put 1,000 in an account that earns 10 percent and it becomes 1,100 after one year, 1,210 after two, and 2,593.74 after ten. The gain keeps getting bigger because each period pays interest on the interest that came before it.

The standard form is y = a(1 + r)^t. The letter a is the starting value, r is the growth rate written as a decimal, and t is the number of periods. A rate of 10 percent enters the formula as 0.10, so the multiplier is 1.10.

Linear growth adds a fixed amount, so the graph is a straight line. Exponential growth multiplies by a fixed factor, so the graph curves upward and the gap between the two widens with every period. The table below runs both side by side from the same 1,000 start.

At period 5 the two columns are close. At period 20 the exponential column is more than twice the linear one, and nothing about the rate changed. That widening gap is the whole reason a small percentage matters over long horizons.

The formula, term by term

Four inputs control the answer, and three of them are easy to mix up.

The period and the rate have to match. A rate quoted per year needs t counted in years. A rate quoted per month needs t counted in months, and the annual figure is then (1 + r)^12 - 1, not 12 x r. At 1 percent a month, the yearly rate is 12.68 percent, because 1.01^12 = 1.1268.

There is a second way to write the same curve, using a continuous rate k: y = a x e^(kt). The two rates are linked by k = ln(1 + r). For r = 0.07, k = 0.0677, and ten years of continuous growth at that rate returns the same 9,835.76 as 1.07^10. Continuous form is convenient in biology and physics; periodic form is what banks and reports use. Mixing them without converting is a common source of small but real errors.

Worked example: 5,000 at 7 percent for 10 years

Take 5,000 invested at 7 percent a year for 10 years. The multiplier is 1.07, applied ten times: 1.07^10 = 1.9672, so the balance is 5,000 x 1.9672 = 9,835.76. The gain is 4,835.76, which is 96.72 percent of the starting amount, not 70 percent. Simple interest at the same rate would pay 5,000 x 0.07 x 10 = 3,500, leaving 8,500.

The gain in year 10 is 643.46, which is larger than the gain in year 1 by 293.46. Nothing improved. The base grew, and 7 percent of a bigger base is a bigger number. Run the same 7 percent for 30 years and 5,000 becomes 38,061.28, with more than half of that total arriving in the last ten years.

Doubling time without guessing

Doubling time answers a practical question: how long until the value is twice as large? Set y = 2a in the formula and the starting value cancels, leaving 2 = (1 + r)^t. Take logs and solve: t = ln(2) / ln(1 + r). At 7 percent that is 0.6931 / 0.0677 = 10.24 years.

The rule of 72 is the mental shortcut: divide 72 by the percentage rate. At 7 percent it gives 10.3 years, within a couple of weeks of the exact 10.24. The shortcut is accurate near 8 percent and drifts at the edges, which the table shows.

For high rates the shortcut undershoots, for low rates it overshoots. Below 1 percent it gets noticeably loose: at 0.5 percent the exact answer is 138.98 years and 72 / 0.5 gives 144. If the number matters, use the log form. The CDC note on epidemic doubling time uses the same relation, doubling time equals ln(2) divided by the growth rate.

Worked example: a colony doubling every 30 minutes

A lab culture starts with 100 cells and doubles every 30 minutes. Six hours is 360 minutes, which is 12 doubling periods, so the count is 100 x 2^12 = 409,600. Written with the periodic formula, the rate per 30 minute period is 100 percent, r = 1.0, and t = 12.

Per hour the count is multiplied by 4, so the hourly rate is 300 percent. The continuous rate is k = ln(2) / 0.5 = 1.386 per hour, and raising e to the 1.386 x 6 power returns the same 409,600. Note how the same colony can be described as growing 100 percent per half hour, 300 percent per hour, or 1.386 per hour continuously. All three are correct, and all three look different in a sentence.

This is why epidemic reports quote a doubling time instead of a percentage. A doubling time does not depend on the length of the reporting window, and it stays constant while growth stays exponential. The US CDC uses the same relation, doubling time equals ln(2) divided by the growth rate, in its published estimates of epidemic doubling times.

Why a straight line chart hides the growth

On a normal axis, exponential growth looks flat for a long time and then explodes. That shape hides the early trend, because the first few periods are compressed against the bottom of the chart. Plot the log of the value instead and a constant growth rate becomes a straight line with slope ln(1 + r).

The slope is the diagnostic. If the logged series is straight, the rate is constant. If it bends down, the rate is falling, which is what a maturing market or a population near its food limit looks like. If it bends up, the rate itself is rising, and no single growth rate describes the whole period.

Fit the rate from two points instead of eyeballing the curve: r = (end / start)^(1/t) - 1. A series that goes from 1,200 to 1,800 in 5 years has r = (1.5)^(1/5) - 1 = 0.0845, or 8.45 percent a year. The same arithmetic is what a compound annual growth rate reports, and it is the honest summary when the year to year numbers jump around.

Solving backwards for the rate or the time

Two reverse questions come up constantly. First, what rate did I get? Second, how long until I reach a target? Both are rearrangements of the same formula.

The reverse formulas assume one constant rate, which is a simplification. A fund that returned 20 percent, then lost 10 percent, then gained 5 percent did not average 5 percent a year. The compounded result is 1.20 x 0.90 x 1.05 = 1.134, or 13.4 percent over the three years, which is 4.28 percent a year. Averaging the three percentages gives 5 percent, and that number is wrong.

Where the model stops working

Exponential growth is a model, and it holds only while the growth rate stays constant. Three things break it in practice.

The first is a limit. Bacteria in a dish, users of a new app and trees in a forest all run into food, attention or space. Growth slows as the quantity approaches the ceiling, and the curve turns into an S shape, the logistic curve. The exponential formula keeps predicting larger numbers past the point where the ceiling should have stopped it, so it is useful for the early phase and misleading near the top.

The second is a changing rate. A 7 percent nominal return with 3 percent inflation delivers 3.88 percent of real purchasing power, because 1.07 / 1.03 = 1.0388. Report the nominal number as nominal. If a chart mixes nominal and real values across decades, the slope is not a growth rate at all.

The third is compounding frequency. A quoted annual rate with monthly compounding produces a higher effective rate than the same quote compounded once a year: 6 percent compounded monthly is 6.17 percent effective, since 1.005^12 = 1.0617. The formula on this page assumes the rate already matches the period you enter. For a rate quoted with a separate compounding schedule, use the compound interest calculator, which handles the frequency directly and shows the APY next to the APR. The US Securities and Exchange Commission's compound interest calculator is a useful cross-check on the same arithmetic.

Enter your own numbers above. The tool returns the final value, the total gain, the multiple, the exact doubling time, the rule of 72 estimate, a period by period table, and, if you supply an end value, the rate that connects the two ends.