Exponential Growth Calculator

For growth that compounds once per period, the final value is x(t) = x0 (1 + r)^t, where x0 is the starting value, r is the rate per period as a decimal and t is the number of periods. For continuous growth it is x(t) = x0 e^rt. Both give a straight line when plotted on a log scale.

Convert the percent rate to a decimal, add 1, raise the result to the power of the number of periods and multiply by the starting value. For 10,000 growing at 7% for 40 periods, 1.07 to the power 40 is 14.97, so the final value is 149,744.58. The calculator returns that value together with the growth factor and the doubling time.

Exponential decay uses the same formulas with a negative rate: x(t) = x0 (1 − r)^t for periodic decay, or x0 e^−kt for continuous decay. A value of 100 shrinking by 10% each period is 72.9 after 3 periods. Enter the rate as a negative number in the calculator.

In growth the value rises by a fixed percentage each period, so the rate is positive and the growth factor is above 1. In decay the value falls by a fixed percentage, so the rate is negative and the factor is between 0 and 1. The same formulas cover both.

Divide the final value by the initial value, take the root of the number of periods and subtract 1. Growing from 100 to 200 in 10 periods gives 2^1/10 − 1 = 7.18% per period. Choose rate as the unknown in the calculator and enter the two values and the time.

Take the natural log of the ratio of final to initial value and divide by the natural log of 1 plus the rate. Doubling at 7% gives ln 2 / ln 1.07 = 10.24 periods. Choose time as the unknown in the calculator and enter the other values.

Divide the final value by the growth factor (1 + r)^t. To end with 50,000 after 15 periods at 6%, divide by 1.06¹⁵ = 2.3966 for an initial value of 20,863.25. Choose initial value as the unknown to run this in reverse.

The Rule of 72 estimates doubling time by dividing 72 by the percentage rate. At 8% it gives 9 periods, close to the true 9.01. It is accurate for low to moderate rates and drifts for high ones, which is why the table above compares it with the exact value from ln 2 / ln(1 + r).

Doubling time is ln 2 divided by ln(1 + r) for periodic growth, or ln 2 divided by r for continuous growth. At 7% per period it is 10.24 periods. The calculator reports it for any positive rate along with the growth factor.

Half-life is the time for a value to fall to half its starting amount. For periodic decay it is ln 0.5 divided by ln(1 + r), where r is negative. At 10% decay per period it is 6.58 periods. The calculator shows it whenever the rate is negative.

Periodic growth multiplies by (1 + r) at the end of each period, while continuous growth multiplies by e^r at every instant. For the same nominal rate, continuous growth gives a slightly larger result. At 3% over 24 periods, 500 grows to 1,027.22 continuously and to 1,016.40 periodically.

The number e is about 2.71828, the base of natural logarithms. It appears in continuous growth, where x(t) = x0 e^kt, because it is the limit of compounding more and more often. The calculator uses it when you choose the continuous model.

The growth factor is the final value divided by the initial value. For periodic growth it equals (1 + r)^t. A growth factor of 14.97 means the value has grown to almost 15 times its start. A factor below 1 means decay.

Divide by 100. A 7% growth rate is 0.07 and the factor per period is 1.07. A 10% decay rate is -0.10 and the factor is 0.90. The calculator takes the percent as you read it and does the division.

Yes. A negative rate means decay, and the calculator treats it that way: 100 falling 10% per period is 72.9 after 3 periods and halves in 6.58 periods. A periodic rate of -100% or less would remove the whole value, so the calculator asks for a rate above -100%.

Yes, for periodic growth. The calculator adds a regular amount at the end of every period using the annuity formula A((1 + r)^t − 1)/r. Adding 100 for 10 periods at 5% from a start of 0 gives 1,257.79. This is available when solving for the final value.

Linear growth adds the same amount each period, while exponential growth multiplies by the same factor, so the amount added keeps growing. Starting at 100 with 10%, linear growth reaches 200 in 10 periods and exponential growth reaches 259.37. The gap widens over time.

Compound interest, early-stage population growth with unlimited resources, unchecked bacterial growth and viral spread are standard examples. Real systems seldom keep a constant rate forever, because limits slow the growth, so the formula describes the phase while the rate is steady.

Radioactive decay, the cooling of a hot object toward room temperature and the fall of a drug's concentration in the body are common examples. Each follows a fixed percentage decline per period or a constant half-life, which the calculator can model.

Use the same formula with the population as the initial value, the growth rate per period and the number of periods. A town of 50,000 growing 2% a year reaches 50,000 × 1.02¹⁰ = 60,950 in 10 years. Choose final value as the unknown and the calculator does the arithmetic.

If a culture doubles every 20 minutes, use a rate of 100% per 20-minute period. After 6 periods, which is 2 hours, a single cell grows to 2⁶ = 64. For a measured doubling time, the rate per period is 2^1/n − 1. Use the rate and time options to check it.

They are the same mathematics. Compound interest is exponential growth where r is the interest rate and t is the number of compounding periods. For the full interest toolkit with deposits and compounding choices, use the compound interest calculator linked below.

Plot time on the horizontal axis and the value on the vertical axis. The curve starts at x0 and rises steeply, with slope proportional to the current value. On a log scale it becomes a straight line whose slope is ln(1 + r). Decay curves fall toward zero without reaching it.

At 5% per period the doubling time is ln 2 / ln 1.05 = 14.21 periods. The Rule of 72 gives 72 / 5 = 14.4. At 3% it is 23.45 periods and at 10% it is 7.27. The doubling table lists the exact times for many rates.

The unit only has to match the rate. A rate of 7% per year goes with time in years, and 7% per month with time in months. If you have a yearly rate and want monthly steps, convert it first, for example with (1 + r)^1/12 − 1, then enter the monthly values.

With a 0% rate the value never changes, so the final value equals the initial value. Solving for time at 0% is impossible, since no time reaches a different final value, and the calculator says so. It still works for the final value, initial value and rate options.

Nearly. Geometric growth is the discrete form: a value multiplied by a fixed ratio each step, so the sequence is x0, x0 r, x0 r squared and so on. Exponential growth is the same idea with time as a continuous variable. The periodic option here gives the geometric sequence.

It uses double-precision arithmetic, and the formulas are on this page so you can check any result by hand. Results are rounded for display. Real-world data seldom fits a constant rate exactly, so treat the output as a model and compare it with your data before relying on it.

Divide the natural log of 2 by the natural log of 1 plus the rate as a decimal. At 3% the doubling time is ln 2 / ln 1.03 = 23.45 periods. The Rule of 72 gives 24. Enter a positive rate and the calculator shows the exact doubling time with every result.

Divide the final value by the initial value, subtract 1 and multiply by 100. Growing from 10,000 to 149,744.58 is a change of 1,397.45%. The calculator reports the growth factor and the total change so you can read the percent directly.

Divide the time by the doubling time. At 7% per period the doubling time is 10.24 periods, so 40 periods holds 3.9 doublings, which matches a growth factor of 2^3.9, close to 15. The doubling time output makes this quick.

Use a negative rate and the half-life output. If a substance loses 10% each period, the half-life is ln 0.5 / ln 0.9 = 6.58 periods. For a known half-life h, the rate per period is 0.5^1/h − 1, which you can enter as the growth rate.

It is the number the starting value is multiplied by. A factor of 2.6533 means the value grew to about 2.65 times its start, and a factor of 0.729 means it fell to 72.9% of the start. It equals (1 + r)^t for periodic growth.

Yes. Enter the inflation rate per year as the rate and the number of years as the time. At 3% a year, a price of 100 becomes 134.39 after 10 years. The same formula applies, although for savings plans the compound interest calculator offers more options.