Key takeaways
- Exponential model: P(t) = P₀ · eᵏᵗ.
- k > 0 means growth; k < 0 means decay.
- Doubling time = ln 2 ÷ k; half-life = ln 2 ÷ |k| (ln 2 ≈ 0.6931).
- Write rates as decimals: 20% is k = 0.2, not 20.
- k and t must use the same time unit.
In this guide
What is exponential growth and decay?
Some things grow by adding the same amount each time: 5, 10, 15, 20. That is linear growth. Others grow by the same percentage: the more there is, the faster it grows. That is exponential growth. The opposite, shrinking by the same percentage, is exponential decay.
Picture bacteria that split in two. With 100 bacteria you get 100 new ones; with 1,000 you get 1,000 new ones. The bigger the colony, the faster it grows. In calculus this is the differential equation dP/dt = kP, and its solution is the exponential formula.
Why does exponential growth matter?
Exponential change fools our instincts. It starts slowly and then seems to take off suddenly. Compare 500 bacteria growing at k = 0.2 per hour with a colony that just adds 100 per hour (the same starting speed).
| Hours | Adds 100 per hour | Exponential, k = 0.2 |
|---|---|---|
| 1 | 600 | 611 |
| 5 | 1,000 | 1,359 |
| 10 | 1,500 | 3,695 |
Same starting speed, but after 10 hours the exponential colony is more than twice as big, and the gap keeps widening.
The formula
| Symbol | What it means | Example |
|---|---|---|
| P(t) | The amount at time t | 1,359.14 bacteria |
| P ₀ | The starting amount at t = 0 (cannot be negative) | 500 |
| k | Continuous rate per unit time: + growth, − decay | 0.2 per hour |
| t | Time, in the same unit as k | 5 hours |
| e | Euler's number | 2.71828 |
The calculator also gives a useful extra: the doubling time ln 2 ÷ k for growth, or the half-life ln 2 ÷ |k| for decay.
How to calculate it step by step
- 1Write down P₀, k and t. Check that k and t use the same time unit.
- 2Multiply k × t. For the example,
0.2 × 5 = 1. - 3Work out e^(kt). Use the exponential calculator:
e¹ ≈ 2.7183. - 4Multiply by P₀.
500 × 2.7183 ≈ 1,359. - 5Optional: doubling time or half-life. Divide 0.6931 by |k|.
Now enter your own starting amount, rate and time below to see the result and the doubling time or half-life.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Using a positive k for decay. Decay needs a negative k, such as −0.1.
- Mixing time units. If k is per year, t must be in years.
- Confusing continuous and yearly compounding.
P₀(1 + r)ᵗis yearly;P₀eᵏᵗis continuous. For money, compare with the compound interest calculator. - Thinking decay reaches zero. In this model it gets closer and closer to 0 but never quite gets there.
Tips and tricks
- Working backwards: to find the time to reach a target, use the natural logarithm:
t = ln(P ÷ P₀) ÷ k. - Finding k from a half-life:
k = −0.6931 ÷ half-life. - Sense check: after one doubling time the amount is 2 × P₀; after two it is 4 × P₀.
Where you'll use it in real life
- Biology: bacteria in a lab and animal populations with plenty of food.
- Medicine: how a drug leaves the body, measured by its half-life.
- Science: radioactive decay and carbon dating.
- Finance: continuously compounded interest and inflation.
Quick summary
When something changes at a rate proportional to its size, P(t) = P₀eᵏᵗ. Positive k grows and negative k decays, and ln 2 ÷ |k| gives the doubling time or half-life. Use the calculator above for the answer with every step shown.
Worked example
A lab starts with 500 bacteria that grow continuously at a rate of k = 0.2 per hour. How many bacteria will there be after 5 hours? Use P₀ = 500, k = 0.2 and t = 5.
- 01ODE: dP/dt = 0.2P (growth)
- 02Solution: P(t) = P₀·e^(kt) = 500·e^(0.2t)
- 03At t = 5: P = 500 × e^(1)
- 04e^(1) = 2.7183
- 05P(5) = 1359.14
- 06Doubling time = ln(2)/k = 3.4657
These numbers are pre-filled in the calculator above.
Frequently asked questions
How do I find the half-life from k?
Half-life = ln(2) ÷ |k|. For example, with k = −0.1 per year, the half-life is about 0.6931 ÷ 0.1 ≈ 6.93 years.
What is the difference between exponential and linear growth?
Linear growth adds the same amount each period. Exponential growth multiplies by the same factor each period, so it starts slowly and then speeds up a lot.
How long does it take to double at 10% continuous growth?
Doubling time = ln(2) ÷ 0.1 ≈ 6.93 time units. A handy shortcut is the rule of 70: 70 ÷ 10 = 7.
What is the doubling time of the bacteria in the example?
With k = 0.2 per hour, the doubling time is 0.6931 ÷ 0.2 ≈ 3.47 hours.
