Exponential growth and decay is one of 1 growth formula in the algebra section of this library, and it is used at high school · ap level.
Why exponential growth and decay works
This is the model for growth that is proportional to the current amount. If each unit of stuff produces the same fraction more per unit of time, the rate of change equals k times the amount present, and e to the kt is the function whose rate of change is k times itself.
What each symbol means
$A_0$ is the initial amount, $k$ the continuous rate, and $t$ time.
Exponential growth and decay: when it holds
Use consistent time units; $k>0$ models growth and $k<0$ decay.
When it stops applying
This model assumes change happens continuously, at every instant. For something that jumps at fixed moments, such as interest paid once a year, it gives a slightly different answer: 1000 at 5 percent grows to 1051.27 continuously but to exactly 1050 with one annual payment.
Exponential growth and decay: a worked example
With $A_0=200$ and $k=0.04$, $A(5)=200e^{0.2}\approx244.28$.
The mistake to avoid
What people do: Entering a percentage rate as a whole number, using k = 4 for a 4 percent rate.
Why it goes wrong: With a starting amount of 200 over 5 years, the correct value is about 244.28, while k = 4 predicts roughly 97 billion. The exponent must be a decimal rate.
Do this instead: Divide the percentage by 100 before it goes anywhere near the exponent, so 4 percent becomes 0.04.
Exponential growth and decay: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $A_0$ is the initial amount, $k$ the continuous rate, and $t$ time.
- Check the conditions before substituting. Use consistent time units; $k>0$ models growth and $k<0$ decay.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most algebra slips.
Where this formula fits
- Subject
- Algebra formulas — 28 entries in this library
- Topic
- Growth
- Level
- High school · AP
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where exponential growth and decay comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Exponentials and Logarithms — the lesson behind this formula: move between growth models and inverse logarithmic form.
- Algebra Calculator — check your substitution and the value it produces.
- Study algebra — the subject guide that explains the ideas these formulas compress.
- Algebra I Practice — questions that make you retrieve the formula instead of recognising it.
- All 28 algebra formulas — the full grouped reference, or the complete formula library.
Questions about exponential growth and decay
How do I find k from a doubling time?
Set the ratio to 2 and solve, which gives k equal to the natural log of 2 divided by the doubling time. A quantity that doubles every 10 years has k of about 0.0693.
How is a half-life handled?
The same way, but with a negative k. Divide the natural log of 2 by the half-life and attach a minus sign, since the amount is falling rather than rising.
Is A equals A0 times (1 + r) to the t the same model?
They describe the same kind of growth in different clothing, and you convert between them because e to the k equals 1 plus r. One counts continuously, the other counts in whole periods.
How do I solve for the time t?
Divide both sides by the starting amount, take the natural log, then divide by k. The log is what pulls t down out of the exponent.