Key takeaways
- f(x) = a × bˣ, where a is the start value and b is the growth factor per step.
- b > 1 means growth; 0 < b < 1 means decay.
- "Grows by 20% each step" means b = 1.2, not b = 0.2.
- Exponential growth accelerates, unlike linear growth which adds a fixed amount.
In this guide
What is an exponential function?
In an exponential function, the variable is in the exponent (the power), not the base. That single difference makes it grow — or shrink — much faster than a straight line, because each step multiplies the previous result instead of adding to it.
Why does exponential growth matter?
Linear growth adds up steadily and predictably. Exponential growth starts small and looks harmless — then suddenly takes off. That surprise is exactly why it matters: things that double repeatedly (population, infections, viral posts) can outgrow expectations extremely fast.
Start with 1 and double it 20 times, and you already have over a million. Double it just 10 more times, and you'd pass a billion. That's the nature of multiplying instead of adding.
The formula
| Symbol | What it means | Example |
|---|---|---|
| f(x) | The value after x steps | 3200 |
| a | The starting value, before any growth | 100 |
| b | The growth factor per step (b > 1 grows, 0 < b < 1 shrinks) | 2 |
| x | The number of steps, often time | 5 |
How to calculate exponential growth step by step
- 1Identify the starting amount, a. This is the value at x = 0.
- 2Identify the growth factor, b. Doubling means b = 2; a 20% increase per step means b = 1.2.
- 3Raise b to the power of x. This is
b^x. - 4Multiply by a.
f(x) = a × b^xis your final answer.
Plug your own starting value, growth factor, and number of steps into the calculator below.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Using a growth rate instead of a growth factor. "Grows by 20% each step" means b = 1.2, not b = 0.2 — you have to add the 1 back in.
- Mixing this up with linear growth. A line adds the same amount each step; an exponential function multiplies. Check the linear equation solver if your quantity is instead growing by a fixed amount.
- Forgetting decay is exponential too. If 0 < b < 1, the value shrinks toward zero rather than growing.
Tips and tricks
- Working backwards from a final value to the number of steps? That's exactly what the logarithm calculator does.
- For money that compounds at a fixed rate over time, see the dedicated compound interest calculator.
- A negative growth factor makes the sign flip every step — rarely useful outside of specific math problems.
Where you'll use it in real life
- Biology: bacteria and cell cultures that double on a fixed schedule.
- Finance: compound interest and investment growth over time.
- Technology: viral shares, where each person spreads content to several more people.
- Science: radioactive decay, where b is less than 1 and the quantity shrinks over time.
Quick summary
Exponential growth multiplies by the same factor at every step instead of adding a fixed amount, which is why it accelerates so fast. Find your starting value and growth factor, raise the factor to the number of steps, then multiply. Try the calculator above to see every step for your own numbers.
Worked example
A lab starts a culture with 100 bacteria. The population doubles every hour. How many bacteria are there after 5 hours?
- 01f(5) = 100 × 2^5
- 02f(5) = 100 × 32
- 03f(5) = 3200
These numbers are pre-filled in the calculator above.
Frequently asked questions
What is the exponential growth formula?
f(x) = a × bˣ, where a is the starting value, b is the growth factor per step, and x is the number of steps.
How do I know if something is growing or decaying?
If the growth factor b is greater than 1, the quantity grows. If b is between 0 and 1, it shrinks (decays) over time.
What's the difference between exponential and linear growth?
Linear growth adds the same fixed amount every step. Exponential growth multiplies by the same factor every step, so it accelerates over time.
Can the growth factor b be negative?
It can, but the result then flips sign every step, which rarely matches a real growth situation — b is almost always positive in practice.
