Project a quantity that grows or shrinks exponentially — population, investment, bacteria or a decaying isotope — and read off the value at any time, the doubling time or half-life, and the curve.
Value at t
1,628.89
Doubling / half-life
14.207
Calculator
Value at t = 10
1,628.89
Growing — multiplied by 1.05 each period.
Per-period factor
1.05
Doubling time
14.207
Total change
628.89
How exponential growth and decay work
A quantity changes exponentially when it is multiplied by the same factor every period. The discrete model uses N(t) = N₀·(1 + r)^t, where r is the growth rate per period (negative for decay). The continuous model uses N(t) = N₀·e^{kt}, where k is the instantaneous rate — the natural choice for things that compound continuously, like radioactive decay.
When the quantity grows, the doubling time ln(2)/ln(base) is how long it takes to double, no matter the starting point. When it decays, the half-life ln(2)/|ln(base)| is how long it takes to halve. Both are constant for a fixed rate, which is the signature of exponential change.
What is the difference between discrete and continuous?
Discrete growth applies the rate once per period (good for annual interest or yearly population figures). Continuous growth compounds at every instant via e^{kt} (good for radioactive decay or continuously compounded interest). For the same headline rate the continuous model grows slightly faster.
How do I model decay?
Enter a negative rate. A discrete rate of −10% gives a per-period factor of 0.9; a continuous rate of −0.1 gives e^−0.1. The calculator then reports a half-life instead of a doubling time.
Why is the doubling time constant?
Because exponential change multiplies by the same factor each period, the time to grow from any value to twice that value is always the same. That fixed doubling time (or half-life) is what distinguishes exponential change from linear change.
Curve
Plot of the exponential curve from t = 0 with the value at the chosen time marked.
Show data table
Time (t)
Value at t
0
1,000
0.25
1,012.27
0.5
1,024.7
0.75
1,037.27
1
1,050
1.25
1,062.89
1.5
1,075.93
1.75
1,089.13
2
1,102.5
2.25
1,116.03
2.5
1,129.73
2.75
1,143.59
3
1,157.63
3.25
1,171.83
3.5
1,186.21
3.75
1,200.77
4
1,215.51
4.25
1,230.42
4.5
1,245.52
4.75
1,260.81
5
1,276.28
5.25
1,291.94
5.5
1,307.8
5.75
1,323.85
6
1,340.1
6.25
1,356.54
6.5
1,373.19
6.75
1,390.04
7
1,407.1
7.25
1,424.37
7.5
1,441.85
7.75
1,459.54
8
1,477.46
8.25
1,495.59
8.5
1,513.94
8.75
1,532.52
9
1,551.33
9.25
1,570.37
9.5
1,589.64
9.75
1,609.15
10
1,628.89
Results are estimates. Verify with a professional for important decisions.
About this calculator
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 results
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.
Worked example
Start with 500 and apply a discrete growth rate of 8% per period for 12 periods.
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.
How it's calculated
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).
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