What this calculator does
An exponential function has the form y = a × bˣ, where a is the starting value at x = 0, b is the base that the value is repeatedly multiplied by, and x is how many steps have passed. It is the standard shape behind compound interest, population growth, radioactive decay and viral spread, anything that changes by a constant multiplier rather than a constant amount.
The base b decides the direction and speed of the change. A base above 1 means growth, a base between 0 and 1 means decay, and a base of exactly 1 means the value never moves at all. This calculator evaluates the function at whatever x you choose and works out the growth or decay rate and the doubling or halving time implied by that base.
The formula
The starting value a is multiplied by the base b raised to the power of x. Because multiplication by a base greater than 1 compounds with every step, and multiplication by a base less than 1 shrinks with every step, small differences in b produce very different results over a large x. The growth rate per step is simply (b − 1) expressed as a percentage.
| Term | Meaning |
|---|---|
| a | The starting value of the function, its value when x = 0. |
| b | The base: the factor the value is multiplied by for each one-unit increase in x. |
| x | The exponent: how many steps or periods have elapsed. |
| Doubling/halving time | How many steps it takes the value to double (b > 1) or halve (b < 1). |
The inputs explained
| Field | What to enter |
|---|---|
| Starting value (a) | The starting value of the function, at x = 0. |
| Base / growth factor (b) | The base. Above 1 for growth, between 0 and 1 for decay. Must be positive. |
| Exponent (x) | The point to evaluate the function at, in whatever units x represents (time periods, generations, and so on). |
When to use it
Projecting a quantity that compounds each period
Anything multiplied by a fixed factor every period, such as an investment compounding at a steady rate or a population growing at a steady percentage, follows this shape exactly.
Modelling decay of a quantity over time
A base below 1 models decay: radioactive material, drug concentration in the body, or the fading effectiveness of a one-off marketing push, each shrinking by the same proportion each period.
Comparing how sensitive an outcome is to the growth rate
Evaluating the same starting value and x at a few different bases shows how much a small change in the underlying rate compounds into a large difference given enough steps.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does the base b change the result after 10 steps?
The same starting value and number of steps, at a range of bases from decay through to growth.
| Base (b) | y value | Growth/decay rate |
|---|---|---|
| 0.9 | 34.868 | 10.00% decay |
| 0.95 | 59.874 | 5.00% decay |
| 1 | 100.000 | 0.000% growth |
| 1.02 | 121.899 | 2.00% growth |
| 1.05 | 162.889 | 5.00% growth |
| 1.1 | 259.374 | 10.0% growth |
| 1.2 | 619.174 | 20.0% growth |
How does the exponent x change the result at a fixed growth rate?
An 8% per-step growth rate, evaluated at a range of x values.
Questions
What is the difference between this and compound interest?
Compound interest is one specific use of this same formula, where a is the principal, b is (1 + interest rate) and x is the number of compounding periods. This calculator works with any exponential relationship, not just money.
What does a base between 0 and 1 mean?
It means decay rather than growth. A base of 0.9, for example, means the value shrinks to 90% of its previous value each step, losing 10% per period compounding downward.
How is this different from the growth-rate calculators for revenue or users?
Those calculators work out a rate from two observed values of a business metric, such as revenue this month versus last month. This one evaluates the function itself at a chosen x once you already know a and b, which is a different, more general calculation.
What happens if the base is zero or negative?
The calculator will not compute a result, because a base of zero collapses to zero for any positive x, and a negative base makes the function alternate sign in a way that is not meaningful for most real-world growth or decay problems.
For solving an exponential equation for x rather than evaluating it, see the exponential equation solver. For roots and other powers, use the exponents and roots calculator.