# What is my exponential function?

Evaluates f(x) = a·bˣ, or finds the exponential function through two points, with the percent growth or decay rate, the continuous rate k = ln b, and the doubling time or half-life.

- Page: https://www.acalculator.org/math/exponential-function-calculator
- JSON spec: https://www.acalculator.org/math/exponential-function-calculator.json
- Version: 3b851b935ff9

## Default answer

Example with the default inputs (I know a and b, a (initial value) 100, b (base) 1.05, x 10): f(x) = 100·1.05ˣ; at your x, f(x) = 162.8894627.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| mode | I know | a and b of f(x) = a·bˣ, or two points the function passes through. |
| a | a (initial value) | f(0), the value where x = 0. It may not be 0. |
| b | b (base) | The growth factor per step of x: more than 0, and not 1. |
| x1 | x₁ | The x of the first point. |
| y1 | y₁ | The y of the first point. Not 0. |
| x2 | x₂ | The x of the second point. Not the same as x₁. |
| y2 | y₂ | The y of the second point: not 0, and the same sign as y₁. |
| x | x | Where to evaluate the function. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| fx | f(x) | a·bˣ at the x you typed. |
| equation | Function | f(x) = a·bˣ with a and b to 10 significant figures. |
| a | a (initial value) | The value of f at x = 0. |
| b | b (growth factor) | The factor f grows by each time x goes up by 1. |
| rate | Rate per step | (b − 1) × 100: positive for growth, negative for decay. |
| kind | Growth or decay | Growth when b > 1, decay when 0 < b < 1. |
| k | Continuous rate k | k = ln b, so f(x) = a·e^(kx). |
| doubling | Doubling time or half-life | ln 2 ÷ \|ln b\|: the change in x that doubles (growth) or halves (decay) f. |

## Method

f(x) = a·bˣ; from two points, b = (y₂ ÷ y₁)^(1 ÷ (x₂ − x₁)) and a = y₁ ÷ b^x₁; rate = (b − 1) × 100%; k = ln b; doubling time or half-life = ln 2 ÷ |ln b|.

## Assumptions

- a ≠ 0, b > 0 and b ≠ 1 (OpenStax’s definition of an exponential function).
- Typed numbers are exact decimals; whole powers are exact, roots and fractional powers are decimal numbers.

## Worked examples

1. mode = points, x1 = -2, y1 = 6, x2 = 2, y2 = 1, x = 0 gives a = 2.44949, b = 0.638943, fx = 2.44949, kind = Exponential decay. Source: OpenStax, Algebra and Trigonometry 2e, §6.1 Exponential Functions, Example 5 (f(x) = 2.4492(0.6389)ˣ through (−2, 6) and (2, 1)), https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-1-exponential-functions.
2. mode = ab, a = 100, b = 1.05, x = 10 gives fx = 162.889463, rate = 5%, kind = Exponential growth, doubling = 14.206699. Source: OpenStax, Algebra and Trigonometry 2e, §6.1 Exponential Functions (f(x) = abˣ, b > 1 is growth), https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-1-exponential-functions.
3. mode = points, x1 = 0, y1 = 3, x2 = 1, y2 = 12, x = 3 gives a = 3, b = 4, fx = 192, rate = 300%. Source: OpenStax, Algebra and Trigonometry 2e, §6.1 Exponential Functions (writing f(x) = abˣ from two points), https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-1-exponential-functions.
4. mode = ab, a = 80, b = 0.5, x = -2 gives fx = 320, doubling = 1, rate = -50%. Source: OpenStax, Algebra and Trigonometry 2e, §6.1 Exponential Functions (0 < b < 1 is decay), https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-1-exponential-functions.

## FAQ

### What is an exponential function?

A function f(x) = a·bˣ, where a is a nonzero number (the initial value, f(0)) and b is a positive number other than 1 (the base). Each time x goes up by 1, f is multiplied by b.

### How do I find an exponential function from two points?

Put both points into y = a·bˣ and divide one equation by the other: y₂ ÷ y₁ = b^(x₂ − x₁), so b = (y₂ ÷ y₁)^(1 ÷ (x₂ − x₁)), then a = y₁ ÷ b^x₁. Through (−2, 6) and (2, 1): b = (1/6)^(1/4) = 0.6389 and a = 6 × 0.6389² = 2.4495, as in OpenStax’s Example 5.

### How can I tell growth from decay?

From b. When b > 1 the function grows; when 0 < b < 1 it decays. The rate per step is (b − 1) × 100%: b = 1.05 grows 5% per step, b = 0.5 shrinks 50% per step.

### What is the continuous rate k?

k = ln b, so the same function is f(x) = a·e^(kx). For b = 1.05, k = 0.04879, a little less than the 5% step rate.

### How do I find the doubling time or the half-life?

Divide ln 2 by |ln b|. For b = 1.05 the doubling time is 0.6931 ÷ 0.04879 = 14.21 steps of x; for b = 0.5 the half-life is exactly 1.

### Why can’t y₁ and y₂ have different signs?

bˣ is always positive, so a·bˣ always has the sign of a. Two points on opposite sides of the x-axis, or a point on it, cannot be on one exponential function.

## Sources

- OpenStax, Algebra and Trigonometry 2e, §6.1 Exponential Functions (f(x) = abˣ with a nonzero and b positive, b ≠ 1; Example 5: through (−2, 6) and (2, 1), f(x) = 2.4492(0.6389)ˣ). https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-1-exponential-functions (retrieved 2026-10-02)