What is the exponential growth formula?

Two formulae, for two different models. Growth applied once per period is A = P (1 + r)^t, so 100 at 10% for 2 periods is 121.00. Growth applied continuously is A = P e^(k t), so the same 100 at the same rate for the same 2 periods is 122.1403. The page substitutes your numbers into whichever formula is on screen and prints the line, so the working is visible rather than implied.

Is 10% growth the same as a continuous rate of 10%?

No, and that is the mistake this page is built to prevent. A growth of 10% each period is the same curve as a continuous rate of 9.531% per period, and a continuous rate of 10% per period is the same curve as a growth of 10.5171% each period. The tool prints the equivalent rate under the answer, so a rate copied from a textbook that uses e can be compared with a rate quoted as a percentage per year.

How do I calculate doubling time?

Divide the natural log of 2 by the growth rate expressed as a log. For growth once per period that is ln 2 / ln(1 + r), which at 10% is 7.2725 periods. For continuous growth it is ln 2 / k, which at the same 10% is 6.9315 periods. The rule of 72 is an approximation of the same thing; this page solves it exactly and shows the figure for the model you picked.

How long does it take to reach a target amount?

Switch Solve for to Time to target and enter the amount you want to reach. From 100 to 250 at 10% each period takes 9.6138 periods, and the same growth applied continuously takes 9.1629 periods. If the rate moves the amount away from the target, or the rate is 0, there is no forward answer and the page says so rather than printing a negative time.

Can it handle decay, or a negative rate?

Yes. Enter a negative rate and the doubling cell becomes a halving cell: 100 at -10% for 2 periods is 81.00 once per period and 81.8731 continuously, halving after 6.5788 periods. Rates of -100% or lower are refused under the discrete model: at -100% the whole amount is gone after one period, and below it the amount would multiply by a negative number, which is not defined between whole periods. The refusal points at the continuous model, where every rate is defined.

What counts as a period?

Whatever you are counting. The tool is unit free: a period is a year for a rate quoted per year, a month for a rent rise, a generation for a colony, a step for anything counted in steps. The rate and the time have to describe the same period, and the results are labelled in periods rather than in any particular unit.

Does this predict what something will be worth?

No. It projects the formula you entered and nothing else. A real population runs into limits, a real price responds to events, and neither follows a constant rate for long. The arithmetic here is exact for the model on screen; whether that model describes the thing you are measuring is the part the page cannot know.

Exponential Growth Calculator

Growth by periods and continuous growth are two different curves, and most calculators show one without saying which they used. This one names the model next to the answer and computes the other model at the same rate beside it: 100 at 10% for two periods is 121.00 applied once per period and 122.1403 applied continuously. It also solves the other direction, the time to reach a target, and prints the doubling or halving time wherever the rate defines one.

Final amount

121.00

100 at 10% for 2 periods · discrete · continuous at the same rate gives 122.1403

Final: 121.00
Model: Discrete
Doubling: 7.2725 periods
Continuous: 122.1403

Starting amount

Growth rate

Rate is

Periods

Model

Solve for

100 at 10% for 2 periods reaches 121.00 under the discrete model. Continuous growth at the same rate gives 122.1403.

121.00 = 100 × (1 + 0.1)^2

A growth of 10% each period is the same curve as a continuous rate of 9.531% per period.

The two models are 1.1403 apart here. e^k is larger than 1 + k for every rate other than 0, so the same rate running without pause always finishes ahead of the same rate applied once each period.

A period is whatever you are counting: a year, a month, a generation, a doubling of a colony. The rate applies to one of them. Results on screen are rounded to four decimal places, and the arithmetic itself keeps full double precision. Nothing you type leaves this browser.

The curve, step by step

Each row is one step along the curve. The bar shows that amount against the largest figure in the table, with the model on screen drawn solid and the other model outlined.

The amount at each step from 0 to 2 periods at 10%, under both models.
Periods	Discrete	Continuous
0	100.00	100.00
1	110.00	110.5171
2	121.00	122.1403

Discrete, the model on screenContinuous, the same rate on the other model

Worked examples

100 at 10%, 2 periodsdiscrete compounding
121.00
100 at rate 0.1, t = 2continuous, 100 e^0.2
122.1403
100 to 250 at 10%solve for time
9.6138 periods

The model on screen. Discrete growth: the amount is multiplied by (1 + r) once each period. The formula is A = P (1 + r)^t. Use it when the growth lands in steps: a yearly rate, a monthly rent rise, a colony counted once a generation.

Accuracy. Exact arithmetic for whichever model you select, and the page always names it, because 10% over two discrete periods gives 121.00 while continuous growth at the same rate gives 122.1403 and neither is a rounding of the other. Doubling and halving time are shown only where the rate makes them defined. It projects the formula you entered and forecasts nothing about a population, a price or an investment.

Common questions

What is the exponential growth formula?: Two formulae, for two different models. Growth applied once per period is A = P (1 + r)^t, so 100 at 10% for 2 periods is 121.00. Growth applied continuously is A = P e^(k t), so the same 100 at the same rate for the same 2 periods is 122.1403. The page substitutes your numbers into whichever formula is on screen and prints the line, so the working is visible rather than implied.
Is 10% growth the same as a continuous rate of 10%?: No, and that is the mistake this page is built to prevent. A growth of 10% each period is the same curve as a continuous rate of 9.531% per period, and a continuous rate of 10% per period is the same curve as a growth of 10.5171% each period. The tool prints the equivalent rate under the answer, so a rate copied from a textbook that uses e can be compared with a rate quoted as a percentage per year.
How do I calculate doubling time?: Divide the natural log of 2 by the growth rate expressed as a log. For growth once per period that is ln 2 / ln(1 + r), which at 10% is 7.2725 periods. For continuous growth it is ln 2 / k, which at the same 10% is 6.9315 periods. The rule of 72 is an approximation of the same thing; this page solves it exactly and shows the figure for the model you picked.
How long does it take to reach a target amount?: Switch Solve for to Time to target and enter the amount you want to reach. From 100 to 250 at 10% each period takes 9.6138 periods, and the same growth applied continuously takes 9.1629 periods. If the rate moves the amount away from the target, or the rate is 0, there is no forward answer and the page says so rather than printing a negative time.
Can it handle decay, or a negative rate?: Yes. Enter a negative rate and the doubling cell becomes a halving cell: 100 at -10% for 2 periods is 81.00 once per period and 81.8731 continuously, halving after 6.5788 periods. Rates of -100% or lower are refused under the discrete model: at -100% the whole amount is gone after one period, and below it the amount would multiply by a negative number, which is not defined between whole periods. The refusal points at the continuous model, where every rate is defined.
What counts as a period?: Whatever you are counting. The tool is unit free: a period is a year for a rate quoted per year, a month for a rent rise, a generation for a colony, a step for anything counted in steps. The rate and the time have to describe the same period, and the results are labelled in periods rather than in any particular unit.
Does this predict what something will be worth?: No. It projects the formula you entered and nothing else. A real population runs into limits, a real price responds to events, and neither follows a constant rate for long. The arithmetic here is exact for the model on screen; whether that model describes the thing you are measuring is the part the page cannot know.

Exact arithmetic for whichever model you select, and the page always names it, because 10% over two discrete periods gives 121.00 while continuous growth at the same rate gives 122.1403 and neither is a rounding of the other. Doubling and halving time are shown only where the rate makes them defined. It projects the formula you entered and forecasts nothing about a population, a price or an investment.