# What is my exponential growth?

Finds the final value of exponential growth or decay from the initial value, the rate per period, and the time, or any one of them from the other three, with the doubling time or half-life.

- Page: https://www.acalculator.org/math/exponential-growth-calculator
- JSON spec: https://www.acalculator.org/math/exponential-growth-calculator.json
- Version: 517663321fc4

## Default answer

Example with the default inputs (Growth Once per period, Find Final value, Initial value (x₀) 100, Growth rate per period (r) 5%, Time (periods) 10): 100 growing at 5% per period (Once per period) for 10 periods becomes 162.889463.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| m | Growth | Compounded once per period, x₀(1 + r)^t, or continuously, x₀e^(rt). |
| find | Find | Which value to work out from the other three. |
| x0 | Initial value (x₀) | The amount at time 0, from 10^-300 to 10^300. |
| r | Growth rate per period (r) | The percent the amount grows each period; negative for decay. More than −100%, at most 1,000%. |
| t | Time (periods) | How many periods pass, such as years or hours, from 0 to 1,000. |
| x | Final value (x(t)) | The amount after t periods, from 10^-300 to 10^300. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| x | Final value (x(t)) | The amount after t periods. |
| x0 | Initial value (x₀) | The amount at time 0. |
| r | Growth rate per period (r) | The percent change per period. |
| t | Time (periods) | How many periods it takes. |
| doubling | Doubling time (periods) | How many periods it takes the amount to double, when the rate is above 0%. |
| halfLife | Half-life (periods) | How many periods it takes the amount to halve, when the rate is below 0%. |
| factor | Growth factor per period | What the amount is multiplied by each period: 1 + r, or e^r for continuous growth. |
| change | Total change | The final value compared with the initial value, as a percent. |

## Method

Once per period: x(t) = x₀ × (1 + r)^t. Continuous: x(t) = x₀ × e^(r × t). Doubling time = ln 2 ÷ ln(1 + r) (continuous: ln 2 ÷ r); half-life = ln 2 ÷ −ln(1 + r) (continuous: ln 2 ÷ −r).

## Assumptions

- r is the percent change per period (5% is 0.05); a negative r is decay. r is more than −100% and at most 1,000%.
- Time is counted in periods and may be a fraction of a period; the rate and the time use the same period.
- The initial and final values are from 10^-300 to 10^300; the time is from 0 to 1,000 periods.

## Worked examples

1. m = discrete, find = x, x0 = 100, r = 5%, t = 10 gives x = 162.889463, doubling = 14.206699, factor = 1.05, change = 62.889463%. Source: Formula from OpenStax College Algebra 2e, section 6.1: https://openstax.org/books/college-algebra-2e/pages/6-1-exponential-functions.
2. m = continuous, find = x, x0 = 100, r = 5%, t = 10 gives x = 164.872127, doubling = 13.862944.
3. m = discrete, find = r, x0 = 1,000, x = 250, t = 2 gives r = -50%, halfLife = 1.
4. m = discrete, find = t, x0 = 500, r = 3%, x = 1,000 gives t = 23.449772.
5. m = discrete, find = x0, r = 10%, t = 3, x = 1,331 gives x0 = 1,000.
6. m = discrete, find = x, x0 = 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, r = -99%, t = 162 gives x = 0.

## FAQ

### What is the exponential growth formula?

x(t) = x₀ × (1 + r)^t, where x₀ is the initial value, r the growth rate per period as a decimal (5% is 0.05), and t the number of periods. $100 growing 5% a period for 10 periods becomes 100 × 1.05^10 ≈ 162.89.

### What is the difference between growth once per period and continuous growth?

Once per period, the amount grows in steps, by r at the end of each period: x₀(1 + r)^t. Continuous growth compounds all the time: x₀e^(rt). At the same r, continuous growth ends a little higher: 100 at 5% for 10 periods gives 162.89 per period and 164.87 continuous.

### How do I calculate exponential decay?

Use a negative rate. A population that shrinks 50% each year has r = −50%: 1000 × (1 − 0.5)² = 250 after 2 years. The rate must be more than −100%, because losing 100% leaves nothing.

### How do I find the growth rate from two values?

Solve the formula for r: r = (x(t) ÷ x₀)^(1/t) − 1 for growth once per period, or r = ln(x(t) ÷ x₀) ÷ t for continuous growth. Pick Rate under Find, then type the initial value, the final value, and the time.

### What is the doubling time?

The time it takes the amount to double: ln 2 ÷ ln(1 + r) periods, or ln 2 ÷ r for continuous growth. At 5% per period it is about 14.2 periods. The rule of 70 (70 ÷ 5 = 14) is a quick estimate of the same number.

### What is a half-life?

For decay, the time it takes the amount to fall by half: ln 2 ÷ ln(1 ÷ (1 + r)) periods. At −50% per period the half-life is exactly 1 period.

### Can the time be a fraction of a period?

Yes. The formula works for any time from 0 up: 2.5 periods at 10% gives x₀ × 1.1^2.5. The rate and the time must use the same period (a yearly rate with time in years).

## Sources

- OpenStax, College Algebra 2e, section 6.1 Exponential Functions (exponential growth and continuous growth): https://openstax.org/books/college-algebra-2e/pages/6-1-exponential-functions
- OpenStax, College Algebra 2e, section 6.7 Exponential and Logarithmic Models (half-life and doubling time): https://openstax.org/books/college-algebra-2e/pages/6-7-exponential-and-logarithmic-models