# Exponential Growth & Decay Calculator — Doubling Time & Half-Life

> Project exponential growth or decay with the continuous N₀·e^{kt} or discrete N₀·(1±r)^t model. Get the value at any time, doubling time or half-life, and a plotted curve.

- **Category:** Math
- **Interactive calculator:** https://youcalc.com/en/math/exponential-growth-decay/
- **Price:** Free, no sign-up required

## Overview

This calculator models exponential growth and decay using either the discrete formula N₀·(1 + r)^t or the continuous formula N₀·e^{kt}. Use it to project population growth, radioactive decay, compound returns, viral spread, or any quantity that multiplies by a fixed factor each period.

## How to read your result

The headline figure is the computed value at the time you entered. Below it, the result card shows the per-period multiplier (base), the doubling time when growing or the half-life when decaying, and the net change from the starting amount. The line chart plots the curve from t = 0 to your chosen time, so you can see how steeply the quantity accelerates or diminishes.

## Method

For the discrete model the value at time t is N(t) = N₀·(1 + r)^t, where N₀ is the starting amount and r is the per-period rate. For the continuous model it is N(t) = N₀·e^{kt}, where k is the continuous growth constant. Both collapse to the unified form N₀·b^t with base b = (1 + r) or b = e^k. Doubling time and half-life are derived from the condition b^T = 2 (or ½), giving T = ln(2) / ln(b).

## Example

- **Setup:** Start with 500 and apply a discrete growth rate of 8% per period for 12 periods.
- **Result:** The ending value is roughly 1,259 — a base multiplier of 1.08 per period, with a doubling time of about 9 periods. The quantity more than doubled from 500 in just 12 steps.

## Frequently asked questions

### When should I use the continuous model instead of the discrete model?

Use the continuous model when growth or decay happens without interruption — for example, radioactive decay, bacterial growth in ideal conditions, or continuously compounded financial returns. Use the discrete model when change occurs in distinct steps, such as annual population counts or period-by-period investment returns.

### What is doubling time, and how is it calculated?

Doubling time is the number of periods needed for the quantity to double. For the discrete model it equals ln(2) / ln(1 + r); for the continuous model it equals ln(2) / k. A higher growth rate means a shorter doubling time — at 10% per period the quantity doubles in roughly 7.3 periods.

### Can I use this for decay, and what is half-life?

Yes. Enter a negative rate for either model and the calculator switches to decay mode. Half-life is the time for the quantity to fall to half its current value. It is computed the same way as doubling time but using the absolute value of the rate: ln(2) / |k| for continuous, or ln(2) / |ln(base)| for discrete.

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## Sources

- https://mathworld.wolfram.com/ExponentialGrowth.html
- https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:exponential-growth-decay

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Interactive version: https://youcalc.com/en/math/exponential-growth-decay/ · From YouCalc — https://youcalc.com