# Exponential Population Growth Calculator

> Project a population with exponential growth, continuous (N₀e^(rt)) or per-period ((1 + r)^t), with the doubling or halving time and a growth curve.

Category: Science & engineering · Applies to: Worldwide · Last updated: 2026-10-29
Interactive calculator: https://calculators.geekmidst.com/calculators/population-growth-exponential-calculator

With unlimited resources a population grows by the same percentage each period, so it doubles at regular intervals. Choose continuous or per-period compounding.

Also called: Exponential growth calculator, Malthusian growth calculator.

## Inputs
- Initial population N₀ (default 1000)
- Growth rate per period (unit %; default 2)
- Number of periods (default 10)
- Model (options: Continuous e^(rt), Per period (1+r)^t; default continuous)

## Results
- Population after t periods
- Change
- Growth factor N ÷ N₀
- Doubling or halving time
- Equivalent continuous rate

## Formula

```
Continuous: N = N₀ × e^(r t);  per period: N = N₀ × (1 + r)^t;  doubling time = ln 2 ÷ k, where k = r or ln(1 + r)
```

Where:
- N₀: initial population
- r: growth rate per period (as a decimal)
- t: number of periods
- k: equivalent continuous growth rate

## Worked example
1. r t = 0.02 × 10 = 0.2.
2. N = 1,000 × e^0.2 = 1,000 × 1.22140 = 1,221.40.
3. Doubling time = ln 2 ÷ 0.02 = 0.693147 ÷ 0.02 = 34.66 periods.

**Result:** A population of 1,000 growing continuously at 2% per period reaches about 1,221 after 10 periods and doubles every 34.7 periods.

## What the result means

Exponential growth describes the early phase of a population with plenty of resources (bacteria in fresh medium, an invasive species). Over longer spans resources run short and growth slows, which the logistic model captures.

## Assumptions and limits
- A constant growth rate (births minus deaths, plus net migration) over the whole period.
- Real populations are limited by resources, so long projections overstate growth. Use the logistic model when a carrying capacity matters.

## Questions

### What is the rule of 70?

Doubling time ≈ 70 ÷ growth rate in %. It comes from ln 2 ≈ 0.693; at 2% it gives 35 periods, close to the exact 34.66.

### Continuous or per period?

Use per period when growth happens in discrete steps (annual breeding seasons); continuous for overlapping generations such as bacteria or human populations.

## Related calculators
- [Logistic Growth Calculator](https://calculators.geekmidst.com/calculators/logistic-growth-calculator)
- [Half-Life Calculator](https://calculators.geekmidst.com/calculators/half-life-calculator)
- [Exponent Calculator](https://calculators.geekmidst.com/calculators/exponent-calculator)

Source: GM¹X CalcVerse — https://calculators.geekmidst.com/calculators/population-growth-exponential-calculator